Solved: In 23-32, write each complex number in rectangular

Plot the point whose rectangular coordinates are (3, - 1) . (p. 156)

To complete the square of x2 + 6x, add __ . (pp. 99-1 OJ)

If P = (a, b) is a point on the terminal side of the angle 0 at a distance rfrom the origin, then sin 0 = __ . (pp. 540-542)

tan-1( -1) = __ . (pp. 608-610)

In polar coordinates, the origin is called the __ and the positive x-axis is referred to as the ____ ____ .

Another representation in polar coordinates for the point ( 2,) is (_, 4;).

The polar coordinates ( -2, ) are represented in rectangular coordinates by ( __ , __ ).

True or False The polar coordinates of a point are unique.

True or False The rectangular coordinates of a point are unique.

True or False In (r, 0), the number r can be negative.

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. (2, _ 1 7T )

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. ( -2, - )

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. (-2,)

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. (2, 7;)

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. 77T 2'6

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. (-2, 5 )

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. -2'6 )

In Problems 11-18, match each point in polar coordinates with either A, B, C, or D on the graph. ( 1 17T 18. 2 - ) 6 -

In Problems 19-30, plot each point given in polar coordinates. (3, 90)

In Problems 19-30, plot each point given in polar coordinates. (4, 270)

In Problems 19-30, plot each point given in polar coordinates. (-2, 0)

In Problems 19-30, plot each point given in polar coordinates. ( -3, 7T)

In Problems 19-30, plot each point given in polar coordinates. ( 6,)

In Problems 19-30, plot each point given in polar coordinates. (5, 5;)

In Problems 19-30, plot each point given in polar coordinates. (-2, 135)

In Problems 19-30, plot each point given in polar coordinates. (-3,120)

In Problems 19-30, plot each point given in polar coordinates. ( - 1, - )

In Problems 19-30, plot each point given in polar coordinates. ( -3, - 3:)

In Problems 19-30, plot each point given in polar coordinates. ( -2, -7T)

In Problems 19-30, plot each point given in polar coordinates. ( -3- . , )

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T ( 5, 2;)

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T (4, 3:)

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T ( -2, 37T)

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T (-3, 47T)

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T (1, 7 2)

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T (2, 7T)

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T -3'-"4 )

In Problems 31-38, plot each point given in polar coordinates, and find other polar coordinates (r, e) of the point for which: (a) r > 0, -27T:S e < 0 (b) r < 0, O:s e < 27T (c) r > 0, 27T:S e < 47T ( 27T -2 ) ' -3

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (3,)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (4, 3;)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (- 2, 0)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( -3, 7T)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (6, 150)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (5, 300)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( 37T -2 -)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( 27T -2 -)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( -1'-)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( _3, _ 3:)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (-2, -180)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( -3, -90)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (7.5, 110)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. ( -3.1, 182)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (6.3, 3.8)

In Problems 39-54, the polar coordinates of a point are gi ven. Find the rectangular coordinates of each point. (8.1, 5.2)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (3, 0)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (0, 2)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (-1, 0)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (0, -2)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (1, -1)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. ( -3, 3)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (V3, 1)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (-2, -2V3)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (1.3, -2.1)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. ( -0.8, -2.1)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. (8.3, 4.2)

In Problems 55-66, the rectangular coordinates of a point are gi ven. Find polar coordinates for each point. ( -2.3, 0.2)

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). 2x2 + 2/ = 3

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). x2 + / = x

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). x2 = 4y

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). / = 2x

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). 2xy = 1

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). 4x2y = 1

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). x = 4

In Problems 67-74, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates (r, e). Y = -3

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r = cos e

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r = sin e + 1

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r2 = cos e

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r = sin e - cos e

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r = 2

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r = 4

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). 4 r=---1 - cos e

In Problems 75-82, the letters r and e represent polar coordinates. Write each equation using rectangular coordinates (x, y). r=--- 3 - cos e

Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with East being the positive x-axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the East direction for the polar axis. Express f) in degrees. (c) U.S. Cellular Field, home of the White Sox, is located at 35th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates. (d) Write the location of U.S. Cellular Field using polar coordinates. Use the East direction for the polar axis. Express f) in degrees.

Show that the formula for the distance d between two points PI = (1'1, f)I) and P2 = h, f)2)is d = VrT + I' - 21'11'2 COS(f)2 - f)1)

In converting from polar coordinates to rectangular coordinates, what formulas will you use?

Explain how you proceed to convert from rectangular coordinates to polar coordinates.

Is the street system in your town based on a rectangular coordinate system, a polar coordinate system, or some other system? Explain.

If the rectangular coordinates of a point are (4, -6), the point symmetric to it with respect to the origin is __ . (pp. 1 67- L 68)

The difference formula for cosine is cos(A - B) = __ . (p. 627)

The standard equation of a circle with center at (-2, 5) and radius 3 is __ . ( pp. 1 89-1 93)

Is the sine function even, odd, or neither? (pp. 556-557)

Slll 4 = __ . (pp. )40-547)

cos 3 = __ . (pp. 540-547)

An equation whose variables are polar coordinates is called a(n) ____ ____.

Using polar coordinates (r, e), the circle x 2 + l = 2x takes the form ____.

A polar equation is symmetric with respect to the pole if an equivalent equation results when r is replaced by __ .

True or False The tests for symmetry in polar coordinates are necessary, but not sufficient.

True or False The graph of a cardioid never passes through the pole.

True or False All polar equations have a symmetric feature.

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r = 4

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r = 2

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. 13 = -3

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. 13 = - - 4

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r sin 13 = 4

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r cos 13 = 4

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r cos 13 = -2

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r sin 13 = -2

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r = 2 cos 13

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r = 2 sin 13

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r = -4 sin 1

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r = -4 cos 13

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r sec 13 = 4

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r csc e = 8

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r csc 13 = -2

In Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r sec 13 = -4

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) r = 2

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) 13 = '!!.. 4

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) r = 2 cos 13

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) r cos 13 = 2

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) r = 1 + cos 13

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) r = 2 sin 13

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) 13 = -4

In Problems 29-36, match each of the graphs (A) through (H) to one of the following polar equations. (A) (B) (C) (D) (E) (F) (G) (H) r sin 13 = 2

In Problems 37-60, identify and graph each polar equation. r = 2 + 2 cos 1

In Problems 37-60, identify and graph each polar equation. r = 1 + sin 1

In Problems 37-60, identify and graph each polar equation. r = 3 - 3 sin 13

In Problems 37-60, identify and graph each polar equation. r = 2 - 2 cos 13

In Problems 37-60, identify and graph each polar equation. r = 2 + sin 13

In Problems 37-60, identify and graph each polar equation. r = 2 - cos 13

In Problems 37-60, identify and graph each polar equation. r = 4 - 2 cos 13

In Problems 37-60, identify and graph each polar equation. r = 4 + 2 sin 1

In Problems 37-60, identify and graph each polar equation. r = 1 + 2 sin 13

In Problems 37-60, identify and graph each polar equation. r = 1 - 2 sin 13

In Problems 37-60, identify and graph each polar equation. r = 2 - 3 cos 1

In Problems 37-60, identify and graph each polar equation. r = 2 + 4 cos 13

In Problems 37-60, identify and graph each polar equation. r = 3 cos(2e)

In Problems 37-60, identify and graph each polar equation. r = 2 sin(3e)

In Problems 37-60, identify and graph each polar equation. r = 4 sin(Se)

In Problems 37-60, identify and graph each polar equation. r = 3 cos( 4(3)

In Problems 37-60, identify and graph each polar equation. r 2 = 9 cos(2e)

In Problems 37-60, identify and graph each polar equation. r 2 = sin(2e)

In Problems 37-60, identify and graph each polar equation. r = 2

In Problems 37-60, identify and graph each polar equation. r = 3

In Problems 37-60, identify and graph each polar equation. r = 1 - cos 13

In Problems 37-60, identify and graph each polar equation. r = 3 + cos 13

In Problems 37-60, identify and graph each polar equation. r = 1 - 3 cos 13

In Problems 37-60, identify and graph each polar equation. r = 4 cos(3e)

In Problems 61-64, the polar equation for each graph is either r = a + b cos 1 or r = a + b sin 1, a > 0, b > 0. Select the correct equation and find the values of a and b. 8 = 5TI 4 2 9 = 3TI

In Problems 61-64, the polar equation for each graph is either r = a + b cos 1 or r = a + b sin 1, a > 0, b > 0. Select the correct equation and find the values of a and b. 8 = 5TI 4 2 9 = 3TI

In Problems 61-64, the polar equation for each graph is either r = a + b cos 1 or r = a + b sin 1, a > 0, b > 0. Select the correct equation and find the values of a and b. 8 = 3TI

In Problems 61-64, the polar equation for each graph is either r = a + b cos 1 or r = a + b sin 1, a > 0, b > 0. Select the correct equation and find the values of a and b. 8 = 3TI

In Problems 65-74, graph each polar equation. r =--- 1 - cos e (parabola)

In Problems 65-74, graph each polar equation. r =---- 1 - 2 cos e (hyperbola)

In Problems 65-74, graph each polar equation. r =---- (ellipse) 3 - 2 cos e

In Problems 65-74, graph each polar equation. r = (parabola) 1 - cos e

In Problems 65-74, graph each polar equation. r = e, e 2: 0 (spiral of Archimedes)

In Problems 65-74, graph each polar equation. r = e (reciprocal spiral)

In Problems 65-74, graph each polar equation. r = csc e - 2, 0 < e < 7T (conchoid)

In Problems 65-74, graph each polar equation. r = sin e tan e (cissoid)

In Problems 65-74, graph each polar equation. r = tan e, 7T 'IT - - < e < - (kappa curve) 2 2

In Problems 65-74, graph each polar equation. r = cos 2

Show that the graph of the equation r sin e = a is a horizontal line a units above the pole if a > 0 and lal units below the pole if a < O.

Show that the graph of the equation r cos e = a is a vertical line a unjts to the right of the pole if a > 0 and lal units to the left of the pole if a < O.

Show that the graph of the equation r = 2a sin e, a > 0, is a circle of radius a with center at (0, a) in rectangular coordinates.

Show that the graph of the equation r = -2a sin e, a > 0, is a circle of radius a with center at (0, -a) in rectangular coordinates.

Show that the graph of the equation r = 2a cos e, a > 0, is a circle of radius a with center at (a, 0) in rectangular coordinates.

Show that the graph of the equation r = -2a cos e, a > 0, is a circle of radius a with center at ( -a, 0) in rectangular coordinates.

Explain why the following test for symmetry is valid: Replace r by -r and 0 by -0 in a polar equation. If an equivalent equation results, the graph is symmetric with respect to the 7T line 0 = "2 (y-axis). (a) Show that the test on page 727 fails for r2 = cos 0, yet this new test works. (b) Show that the test on page 727 works for r2 = sin 0, yet this new test fails.

Develop a new test for symmetry with respect to the pole. (a) Find a polar equation for which this new test fails, yet the test on page 727 works. (a) Show that the test on page 727 fails for r2 = cos 0, yet this new test works. (b) Find a polar equation for which the test on page 727 fails, yet the new test works.

Write down two different tests for symmetry with respect to the polar axis. Find examples in which one test works and the other fails. Which test do you prefer to use? Justify your answer.

The conjugate of -4 - 3i is ___ . (pp. 109-114)

The sum formula for the sine function is sin (A + B) = ____. (p. 630)

The __ sum formula for the cosine function is cos (A + 8) = ____ . (p. 627)

sin 120 = __ ; cos 240 = __ . (pp. 540-547)

When a complex number z is written in the polar form z = r(cos e + i sin e), the nonnegative number r is the __ or __ of z, and the angle e, 0 s; e < 271", is the __ of z.

___ Theorem can be used to raise a complex number to a power.

Every non-zero complex n umber will have exactly __ cube roots.

True or False De Moivre's Theorem is useful for raising a complex number to a positive integer power.

True or False Using De Moivre's Theorem, the square of a complex number will have two answers.

True or False The polar form of a complex number is unique.

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. 1 + i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. -1 + i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. V3 - i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. 1 - V3i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. -3i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. -2

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. 4 - 4i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. 9V3 + 9i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. 3 - 4i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. 2+ V3i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. -2 + 3i

In Problems 11-22, plot each complex nu.mber in the complex plane and write it in polar form. Express the argument in degrees. Vs - i

In Problems 23-32, write each complex number in rectangular form. 2( cos 120 + i sin 120)

In Problems 23-32, write each complex number in rectangular form. 3( cos 210 + i sin 210)

In Problems 23-32, write each complex number in rectangular form. 4 cos -=! + i sin 4

In Problems 23-32, write each complex number in rectangular form. 2 ( cos 6 57T . + i Sin 6 57T)

In Problems 23-32, write each complex number in rectangular form. 3 cos T + i sin T 7T)

In Problems 23-32, write each complex number in rectangular form. 4 ( cos "2 7T + i sin "2 7T)

In Problems 23-32, write each complex number in rectangular form. 0.2( cos 100 + i sin 100)

In Problems 23-32, write each complex number in rectangular form. 0.4( cos 200 + i sin 200)

In Problems 23-32, write each complex number in rectangular form. 2 ( cos 7T . 7T) 18 + i Sin 1

In Problems 23-32, write each complex number in rectangular form. 3 ( cos 7T 7T) 10 + i sin

In Problems 33-40, find zw and . Leave your answers in polar form. z = 2( cos 40 + i sin 40) w = 4( cos 20 + i sin 20)

In Problems 33-40, find zw and . Leave your answers in polar form. z = cos 120 + i sin 120 w = cos 100 + i sin 100

In Problems 33-40, find zw and . Leave your answers in polar form. z = 3(cos 130 + i sin 1300) w = 4( cos 270 + i sin 270)

In Problems 33-40, find zw and . Leave your answers in polar form. z = 2( cos 80 + i sin 80) w = 6( cos 200 + i sin 200)

In Problems 33-40, find zw and . Leave your answers in polar form. z = 2( cos i + i sin i) W = 2 ( 7T cos - + i sin -7T ) 10 10

In Problems 33-40, find zw and . Leave your answers in polar form. Z = 4 cos 8 + i sin 8 7T) ( 97T 97T) W = 2 cos 16 + i sin 1

In Problems 33-40, find zw and . Leave your answers in polar form. z = 2 + 2i w=V3-i

In Problems 33-40, find zw and . Leave your answers in polar form. z = 1 - i w = 1 - V3i

In Problems 41-52, write each expression in the standard form a + bi.

In Problems 41-52, write each expression in the standard form a + bi. [3( cos 80 + i sin 80) ]3

In Problems 41-52, write each expression in the standard form a + bi. 2 cos ) J5 10 + i sin 10

In Problems 41-52, write each expression in the standard form a + bi. [ V2 (cos 57 + i sin 57)]4

In Problems 41-52, write each expression in the standard form a + bi. [ V3 (cos 10 + i sin 10)]6

In Problems 41-52, write each expression in the standard form a + bi. [ 1 ( cos 72 + i sin 72)]5

In Problems 41-52, write each expression in the standard form a + bi. [ , ;:: (37T . 37 T ) ]4 v5 cos 16 + i Sin 1

In Problems 41-52, write each expression in the standard form a + bi. [ , ;:: ( 57T . 57 T ) ]6

In Problems 41-52, write each expression in the standard form a + bi. (1 - i)5

In Problems 41-52, write each expression in the standard form a + bi. ( V3 - i )6

In Problems 41-52, write each expression in the standard form a + bi. ( V2 - i )6

In Problems 41-52, write each expression in the standard form a + bi. ( 1 - V5i )8

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 1 + i

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of V3 - i

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of 4 - 4 V3i

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of -8 - 8i

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of -16i

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of -8

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fifth roots of i

In Problems 53-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fifth roots of -i

Find the four complex fourth roots of unity (1) and plot them.

Find the six complex sixth roots of unity (1) and plot them.

Show that each complex nth root of a nonzero complex number w has the same magnitude.

Use the result of Problem 63 to draw the conclusion that each complex nth root lies on a circle with center at the origin. What is the radius of this circle?

Refer to Problem 64. Show that the complex nth roots of a nonzero complex number w are equally spaced on the circle.

Prove formula (6).

A vector whose magnitude is 1 is called a(n) __ vector.

The pro duct of a vector by a number is called a(n) __ multiple.

If v = ai + bj , then a is called the __ component of v and b is the __ component of v.

True or False Vectors are quantities that have magnitude and direction.

True or False Force is a physical example of a vector.

True or False Mass is a physical example of a vector.

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V v + w

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V u + v

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V 3v

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V 4w

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V v - w

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V u - v

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V 3v + u - 2w

In Problems 7-14, use the vectors in the figure at the right to graph each of the following vectors. ! u I V 2u - 3v + w

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. A + B = F

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. K + G = F

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. C = D - E + F

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. G + H + E = D

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. E + D = G + H

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. H - C = G - F

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. A + B + K + G = 0

In Problems 15-22, use the figure at the right. Determine whelher the given statement is true or false. A + B + C + H + G = 0

If Ilvll = 4, what is 113vll?

If I lvll = 2, what is 11-4vll?

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = (0, 0); Q = (3, 4)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = (0, 0); Q = ( -3, -5)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = (3, 2); Q = (5, 6)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = ( -3, 2); Q = ( 6, 5)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = ( -2, -1); Q = (6, -2)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = ( - 1 , 4); Q = (6, 2)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = (1, 0); Q = (0, 1)

In Problems 25-32, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. P = (1, 1); Q = (2, 2)

In Problems 33-38, find Ilvll . v = 3i - 4j

In Problems 33-38, find Ilvll . v = -5i + 1 2j

In Problems 33-38, find Ilvll . v = i - j

In Problems 33-38, find Ilvll . v = -i - j

In Problems 33-38, find Ilvll . v = -2i + 3j

In Problems 33-38, find Ilvll . v = 6i + 2j

In Problems 39-44,find each quantity if v = 3i - 5j and w = -2i + 3j. 2v + 3w

In Problems 39-44,find each quantity if v = 3i - 5j and w = -2i + 3j. 3v - 2w

In Problems 39-44,find each quantity if v = 3i - 5j and w = -2i + 3j. Ilv - wll

In Problems 39-44,find each quantity if v = 3i - 5j and w = -2i + 3j. Ilv + wl l

In Problems 39-44,find each quantity if v = 3i - 5j and w = -2i + 3j. Ilvll - Ilwll

In Problems 39-44,find each quantity if v = 3i - 5j and w = -2i + 3j. Ilvl l + Ilwll

In Problems 45-50, find the unit vector in the same direction as v. v = 5i

In Problems 45-50, find the unit vector in the same direction as v. v = -3j

In Problems 45-50, find the unit vector in the same direction as v. v = 3i - 4j

In Problems 45-50, find the unit vector in the same direction as v. v = -5i + 1 2j

In Problems 45-50, find the unit vector in the same direction as v. v = i - j

In Problems 45-50, find the unit vector in the same direction as v. v = 2i - j

Find a vector v whose magnitude is 4 and whose component in the i direction is twice the component in the j direction.

Find a vector v whose magnitude is 3 and whose component in the i direction is equal to the component in the j direction.

If v = 2i - j and w = xi + 3j, find all numbers x for which Ilv + wll = 5.

If P = (-3, 1) and Q = (x, 4), find all numbers x sllch that the vector represented by PQ has length 5.

In Problems 55-60, write the vector v in (he form ai + bj, given its magnitude Ilvl l and the angle a it makes with the positive x-axis. Ilvll = 5, a = 60

In Problems 55-60, write the vector v in (he form ai + bj, given its magnitude Ilvl l and the angle a it makes with the positive x-axis. Ilvll = 8, a = 45

In Problems 55-60, write the vector v in (he form ai + bj, given its magnitude Ilvl l and the angle a it makes with the positive x-axis. Ilvl l = 14, a = 120

In Problems 55-60, write the vector v in (he form ai + bj, given its magnitude Ilvl l and the angle a it makes with the positive x-axis. Ilvl l = 3, a = 240

In Problems 55-60, write the vector v in (he form ai + bj, given its magnitude Ilvl l and the angle a it makes with the positive x-axis. Ilvll = 25, a = 330

In Problems 55-60, write the vector v in (he form ai + bj, given its magnitude Ilvl l and the angle a it makes with the positive x-axis. Ilvll = 15, a = 315

Computer Graphics The field of computer graphics utilizes vectors to compute translations of points. For example, if the point ( -3, 2) is to be translated by v = (5, 2), then the new location will be u' = u + v = ( -3, 2) + (5, 2) = (2, 4). As illustrated in the figure, the point ( -3, 2) is translated to (2, 4) by v. Source: Phil Dadd. Vectors and Matrices: A Primer. www.gamedev.netlreferencelarticleslarticle1832.asfJ (a) Determine the new coordinates of (3, -1 ) if it is translated by v = ( -4, 5) (b) Illustrate this translation graphically. y (-3, 2) -5 x

Computer Graphics Refer to Problem 61 . The points ( -3, 0), (-1, -2), (3, 1) and (1, 3) are the vertices of a parallelogram A BeD. (a) Find the new vertices of a parallelogram A'B'C'D' if it is translated by v = (3, -2). (b) Find the new vertices of a parallelogram A' B'C' D' if it . 1 IS translated by - "

Force Vectors A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of 30 with the ground. Express the force vector F in terms of i and j.

Force Vectors A man pushes a wheelbarrow up an incline of 20 wi th a force of 100 pounds. Express the force vector F in terms of i and j.

Resultant Force Two forces of magnitude 40 newtons (N) and 60 N act on an object at angles of 30 and -45 with the positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find F\ + F2 .

Resultant Force Two forces of magnitude 30 newtons (N) and 70 N act on an object at angles of 45 and 120 with the positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find F\ + F2

Static Equilibrium A weight of 1000 pounds is suspended from two cables as shown in the figure. What are the tensions in the two cables?

Static Equilibrium A weight of 800 pounds is suspended from two cables, as shown in the figure. What are the tensions in the two cables?

Static Equilibrium A tightrope walker located at a certain point deflects the rope as indicated in the figure. If the weight of the tightrope walker is 150 pounds, how much tension is in each part of the rope? 3.r 1 50 pounds

Static Equilibrium Repeat Problem 69 if the left angle is 3.8, the right angle is 2.6, and the weight of the tightrope walker is 135 pounds.

Trnck Pull At a county fair truck pull, two pickup trucks are attached to the back end of a monster truck as illustrated in the figure. One of the pickups pulls with a force of 2000 pounds and the other pulls with a force of 3000 pounds with an angle of 45 between them. With how much force must the monster truck pull in order to remain unmoved? (Hint: Find the resultant force of the two trucks.) ..--A--. f0'0 45 J 3000 lb

Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor's tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a 40 angle between the forces as shown in the figure. (a) Assuming the farmer's estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump? Explain. (b) Had the farmer arranged the tractors with a 25 angle between the forces, would he have been successful in removing the stump? Explain.

Static Equilibrium Show on the following graph the force needed for the object at P to be in static equilibrium.

Explain in your own words what a vector is. Give an example of a vector.

Write a brief paragraph comparing thc algebra of complex numbers and the algebra of vectors.

In a triangle with sides a, b, c and angles A, B, C, the Law of Cosines states that ___ . (p. 686)

If v w = 0, then the two vectors v and w are __

If v = 3w, then the two vectors v and w are __ .

True or False If v and w are parallel vectors, then v . w = O.

True or False Given two nonzero vectors v and w, it is always possible to decompose v into two vectors, one parallel to w and the other perpendicular to w.

True or False Work is a physical example of a vector.

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = i - j, w = i + j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = i + j, w = -i + j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = 2i + j, w = i - 2j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = 2i + 2j, w = i + 2j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = V3i - j, w = i + j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = i + V3j, w = i - j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = 3i + 4j, w = 4i + 3j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = 3i - 4j, w = 4i - 3j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = 4i, w = j

In Problems 7-16, (a) find the dot product V w; (b) find the angle between v and w; (c) state whether the vectors are parallel, orthogonal, or neither. v = i, w = -3j

Find a so that the vectors v = i - aj and w = 2i + 3j are orthogonal.

Find b so that the vectors v = i + j and w = i + bj are orthogonal.

In Problems 19-24, decompose v into two vectors VI and V2 , where VI is parallel to w and V2 is orthogonal to w. v = 2i - 3j, w = i - j

In Problems 19-24, decompose v into two vectors VI and V2 , where VI is parallel to w and V2 is orthogonal to w. v = -3i + 2j, w = 2i + j

In Problems 19-24, decompose v into two vectors VI and V2 , where VI is parallel to w and V2 is orthogonal to w. v = i - j, w = -i - 2j

In Problems 19-24, decompose v into two vectors VI and V2 , where VI is parallel to w and V2 is orthogonal to w. v = 2i - j, w = i - 2j

In Problems 19-24, decompose v into two vectors VI and V2 , where VI is parallel to w and V2 is orthogonal to w. v = 3i + j, w = -2i - j

In Problems 19-24, decompose v into two vectors VI and V2 , where VI is parallel to w and V2 is orthogonal to w. v = i - 3j, w = 4i - j

Computing Work Find the work done by a force of 3 pounds acting in the direction 60 to the horizontal in moving an object 6 feet from (0, 0) to (6, 0).

Computing Work A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of 30 with the horizontal. How much work is done in moving the wagon 1 00 feet?

Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun's rays and the area of the panel. Let the vector I represent the intensity, in watt per square centimeter, having the direction of the sun's rays. Let the vector A represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure. The total number of watts collected by the panel is given by W = II AI. Suppose I = ( -0.02, -0.01) and A = (300, 400). (a) Find 11I11 and IIAII and interpret the meaning of each. (b) Compute W and interpret its meaning. (c) If the solar panel is to collect the maximum number of watts, what must be true about I and A?

Rainfall Measurement Let the vector R represent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector A represent the area, in square inches, whose direction is the orientation of the opening of the rain gauge. See the figure. The volume of rain collected in the gauge, in cubic inches, is given by V = I R AI, even when the rain falls in a slanted direction or the gauge is not perfectly vertical. . 29. Suppose R = (0.75, - 1.75) and A = (0.3, 1). R\\\\ A 9 - 8 - 7 6 - 5 4 - 3 - 2 - 1 (a) Find \\R\\ and IIAII and interpret the meaning of each. (b) Compute V and interpret its meaning (c) If the gauge is to collect the maximum volume of rain, what must be true about R and A?

Finding the Actual Speed and Direction of an Aircraft A Boeing 747 jumbo jet maintains an airspeed of 550 miles per hour in a southwesterly direction. The velocity of the jet stream is a constant 80 miles per hour from the west. Find the actual speed and direction of the aircraft. -- -- -- , Jet stream

Finding the Correct Compass Heading The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 miles per hour from the northwest. If the pilot maintains an airspeed of 250 miles per hour, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft?

Correct Direction for Crossing a River A river has a constant current of 3 kilometers per hour. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kilometers per hour, be headed in order to reach a point directly opposite the dock? If the river is 1:. kilometer wide, how long will it take to cross? 2 Current -- 1 Direction of boat due to current

Crossing a River A small motorboat in still water maintains a speed of 20 miles per hour. In heading directly across a river (that is, perpendicular to the current) whose current is 3 miles per hour, find a vector representing the speed and direction of the motorboat. What is the true speed of the motorboat? What is its direction?

Braking Load A Toyota Sienna with a gross weight of 5300 pounds is parked on a street with a slope of 8. See the figure. Find the force required to keep the Sienna from rolling down the hill. What is the force perpendicular to the hill? Weight = 5300 pounds

Braking Load A Pontiac Bonneville with a gross weight of 4500 pounds is parked on a street with a slope of 10. Find the force required to keep the Bonneville from rolling down the hill. What is the force perpendicular to the hill?

Grollnd Speed and Direction of an Aiq)lane An airplane has an airspeed of 500 kilometers per hour (km/hr) bearing N45E. The wind velocity is 60 km/hr in the direction N30W. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

Ground Speed and D irection of an Airplane An airplane has an airspeed of 600 km/hr bearing S300E. The wind velocity is 40 km/hr in the direction S45E. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

Ramp Angle Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full with other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is 20, how many pounds of force must Timmy exert to hold the piano in position? 250lb

Incline Angle A bulldozer exerts 1000 pounds of force to prevent a SOOO-pound boulder from rolling down a hill. Determine the angle of inclination of the hill.

Find the acute angle that a constant unit force vector makes with the positive x-axis if the work done by the force in moving a particle from (0, 0) to (4,0) equals 2.

Prove the distributive property: u'(v + w) = u'V + u'W

Prove property (S), 0 0 v = O.

If v is a unit vector and the angle between v and i is 0', show that v = cos O'i + sin O'j.

Suppose that v and ware unit vectors. If the angle between v and i is 0' and that between wand i is {3, use the idea of the dot product vow to prove that cos(O' - {3) = cos 0' cos {3 + sin 0' sin {3

Show that the projection of v onto i is (v i)i. Then show that we can always write a vector v as v = (voi)i + (v'j)j

(a) If u and v have the same magnitude, show that 1I + vand u - v are orthogonal. (b) Use this to prove that an angle inscribed in a semicircle is a right angle (see the figure).

Let v and w denote two nonzero vectors. Show that the vecV'W tor v - O'W is orthogonal to w if 0' = Ilw112'

Let v and w denote two nonzero vectors. Show that the vectors Ilwllv + Ilvllw and Ilwllv - Ilvllw are orthogonal.

In the definition of work given in this section, what is the work done if F is orthogonal to AB?

Prove the polarization identity, Ilu + vl12 - Ilu - vl12 = 4(u' v)

Create an application different from any found in the text that requires a dot product.

In Problems 1-6, plot each point given in polar coordinates, and find its rectangular coordinates. ( 3-7 6 . , )

In Problems 1-6, plot each point given in polar coordinates, and find its rectangular coordinates. 27i 4'3

In Problems 1-6, plot each point given in polar coordinates, and find its rectangular coordinates. ( -2-471 . , ) 3

In Problems 1-6, plot each point given in polar coordinates, and find its rectangular coordinates. ( -1 571 . , )

In Problems 1-6, plot each point given in polar coordinates, and find its rectangular coordinates. ( -3 , -) 2

In Problems 1-6, plot each point given in polar coordinates, and find its rectangular coordinates. ( -4 - ) ' 4

In Problems 7-12, the rectangular coordinates of a point are given. Find two pairs of polar coordinates (r, (J) for each point, one with r > 0 and the other with r < O. Express (J in radians. ( -3, 3)

In Problems 7-12, the rectangular coordinates of a point are given. Find two pairs of polar coordinates (r, (J) for each point, one with r > 0 and the other with r < O. Express (J in radians. (1, -1)

In Problems 7-12, the rectangular coordinates of a point are given. Find two pairs of polar coordinates (r, (J) for each point, one with r > 0 and the other with r < O. Express (J in radians. (0, -2)

In Problems 7-12, the rectangular coordinates of a point are given. Find two pairs of polar coordinates (r, (J) for each point, one with r > 0 and the other with r < O. Express (J in radians. (2, 0)

In Problems 7-12, the rectangular coordinates of a point are given. Find two pairs of polar coordinates (r, (J) for each point, one with r > 0 and the other with r < O. Express (J in radians. (3, 4)

In Problems 7-12, the rectangular coordinates of a point are given. Find two pairs of polar coordinates (r, (J) for each point, one with r > 0 and the other with r < O. Express (J in radians. ( -5, 12)

In Problems 13-18, the letters r and (J represent polar coordinates. (a) Write each polar equation as an equation in rectangular cOO/'dinates (x, y). (b) identify the equation and graph it. r = 2 sin (J

In Problems 13-18, the letters r and (J represent polar coordinates. (a) Write each polar equation as an equation in rectangular cOO/'dinates (x, y). (b) identify the equation and graph it. 3r = sin (J

In Problems 13-18, the letters r and (J represent polar coordinates. (a) Write each polar equation as an equation in rectangular cOO/'dinates (x, y). (b) identify the equation and graph it. r = 5

In Problems 13-18, the letters r and (J represent polar coordinates. (a) Write each polar equation as an equation in rectangular cOO/'dinates (x, y). (b) identify the equation and graph it. (J = . 4

In Problems 13-18, the letters r and (J represent polar coordinates. (a) Write each polar equation as an equation in rectangular cOO/'dinates (x, y). (b) identify the equation and graph it. r cos (J + 3r sin (J = 6

In Problems 13-18, the letters r and (J represent polar coordinates. (a) Write each polar equation as an equation in rectangular cOO/'dinates (x, y). (b) identify the equation and graph it. r 2 + 4r sin (J - 8r cos (J = 5

In Problems 19-24, sketch the graph of each polar equation. Be sure to test for symmetry. r = 4 cos (J

In Problems 19-24, sketch the graph of each polar equation. Be sure to test for symmetry. r = 3 sin (

In Problems 19-24, sketch the graph of each polar equation. Be sure to test for symmetry. r = 3 - 3 sin (J

In Problems 19-24, sketch the graph of each polar equation. Be sure to test for symmetry. r = 2 + cos (J

In Problems 19-24, sketch the graph of each polar equation. Be sure to test for symmetry. r = 4 - cos (J

In Problems 19-24, sketch the graph of each polar equation. Be sure to test for symmetry. r = 1 - 2 sin (J

In Problems 25-28, write each complex number in polar form. Express each argument in degrees . -1 - i

In Problems 25-28, write each complex number in polar form. Express each argument in degrees . -v3 + i

In Problems 25-28, write each complex number in polar form. Express each argument in degrees . 4 - 3i

In Problems 25-28, write each complex number in polar form. Express each argument in degrees . 3 - 2i

In Problems 29-34, write each complex number in the standard form a + bi and plot each in the complex plane. 2( cos 150 + i sin 150)

In Problems 29-34, write each complex number in the standard form a + bi and plot each in the complex plane. 3( cos 60 + i sin 60)

In Problems 29-34, write each complex number in the standard form a + bi and plot each in the complex plane. 3 cos 3" + i SIn 3" )

In Problems 29-34, write each complex number in the standard form a + bi and plot each in the complex plane. 4 cos 4 + isin4

In Problems 29-34, write each complex number in the standard form a + bi and plot each in the complex plane. 0.1(cos 350 + i sin 350)

In Problems 29-34, write each complex number in the standard form a + bi and plot each in the complex plane. 0.5(cos160 + i sin 160)

In Problems 35-40, find zw and . Leave your answers in polar form. z = cos 80 + i sin 80 w = cos 50 + i sin 50

In Problems 35-40, find zw and . Leave your answers in polar form. z = cos 205 + i sin 205 w = cos 85 + i sin 85

In Problems 35-40, find zw and . Leave your answers in polar form. z = 3 coss + ism. 917) S ( 17 17 W = 2 cos 5 + i sin 5 )

In Problems 35-40, find zw and . Leave your answers in polar form. z = 2( cos 5 ; + i sin 5;) w = 3(COS + iSin)

In Problems 35-40, find zw and . Leave your answers in polar form. z = 5( cos 10 + i sin 10) w = cos 355 + i sin 355

In Problems 35-40, find zw and . Leave your answers in polar form. z = 4( cos 50 + i sin 50) w = cos 340 + i sin 340

In Problems 41-48, write each expression in the standard form a + bi. [3( cos 20 + i sin 20) J3

In Problems 41-48, write each expression in the standard form a + bi. [2( cos 50 + i sin 50) J3

In Problems 41-48, write each expression in the standard form a + bi. [, ;;:: (517 517) ]4 V 2 cos 8 + i sin 8

In Problems 41-48, write each expression in the standard form a + bi. [2( cos + i sin ) r

In Problems 41-48, write each expression in the standard form a + bi. (1 - V3i)6

In Problems 41-48, write each expression in the standard form a + bi. (2 - 2i)8

In Problems 41-48, write each expression in the standard form a + bi. (3 + 4i)4

In Problems 41-48, write each expression in the standard form a + bi. (1 - 2i)4

Find all the complex cube roots of 27.

Find all the complex fourth roots of -16.

In Problems 51-54, use the figure to graph each of the following: u + v

In Problems 51-54, use the figure to graph each of the following: v + w

In Problems 51-54, use the figure to graph each of the following: 2u + 3v

In Problems 51-54, use the figure to graph each of the following: 5v - 2 w

In Problems 55-58, the vector v is represented by the directed line segment PQ. Write v in the form ai + bj and find Ilvll. P = (1, -2); Q = (3, -6)

In Problems 55-58, the vector v is represented by the directed line segment PQ. Write v in the form ai + bj and find Ilvll. P = (-3,1); Q = (4, -2)

In Problems 55-58, the vector v is represented by the directed line segment PQ. Write v in the form ai + bj and find Ilvll. P = (0, -2); Q = (-1,1)

In Problems 55-58, the vector v is represented by the directed line segment PQ. Write v in the form ai + bj and find Ilvll. P = (3, -4); Q = (-2, 0)

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: v + w

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: v - w

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: 4v - 3w

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: -v + 2 w

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: II vii

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: I v + l wll

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: Ilvll + Ilwll

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: 12vll - 311wll

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: Find a unit vector in the same direction as v.

In Problems 59-68, use the vectors v = -2i + j and w = 4i - 3j to find: Find a unit vector in the opposite direction of w.

Find the vector v in the xy-plane with magnitude 3 if the angle between v and i is 60.

Find the vector v in the xy-plane with magnitude 5 if the angle between v and i is 150.

In Problems 71- 74, find the dot product v w and the angle between v and w. v = -2i + j, w = 4i - 3j

In Problems 71- 74, find the dot product v w and the angle between v and w. v = 3i - j, w = i + j

In Problems 71- 74, find the dot product v w and the angle between v and w. v = i - 3j, w = -i + j

In Problems 71- 74, find the dot product v w and the angle between v and w. v = i + 4j, w = 3i - 2j

In Problems 75-80, determine whether v and w are parallel, orthogonal, or neither. v = 2i + 3j; w = -4i - 6j

In Problems 75-80, determine whether v and w are parallel, orthogonal, or neither. v = -2i - j; w = 2i + j

In Problems 75-80, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 4j; w = -3i + 4j

In Problems 75-80, determine whether v and w are parallel, orthogonal, or neither. v = -2i + 2j; w = -3i + 2j

In Problems 75-80, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 2j; w = 4i + 6j

In Problems 75-80, determine whether v and w are parallel, orthogonal, or neither. v = -4i + 2j; w = 2i + 4j

In Problems 81-82, decompose v into two vectors, one parallel to w and the other orthogonal to w. v = 2i + j; w = -4i + 3j

In Problems 81-82, decompose v into two vectors, one parallel to wand the other orthogonal to w. v = -3i + 2j; w = -2i + j

Actual Speed and Direction of a Swimmer A swimmer can maintain a constant speed of 5 miles per hour. If the swimmer heads directly across a river that has a current moving at the rate of 2 miles per hour, what is the actual speed of the swimmer? (See the figure.) If the river is 1 mile wide, how far downstream will the swimmer end up from the point directly across the river from the starting point?

Actual Speed and Direction of a Swimmer A swimmer can maintain a constant speed of 5 miles per hour. If the swimmer heads directly across a river that has a current moving at the rate of 2 miles per hour, what is the actual speed of the swimmer? (See the figure.) If the river is 1 mile wide, how far downstream will the swimmer end up from the point directly across the river from the starting point? easterly direction. Find the actual speed and direction of the plane relative to the ground.

Static Equilibrium A weight of 2000 pounds is suspended from two cables, as shown in the figure. What are the tensions in each cable?

Actual Speed and Distance of a Motorboat A small motorboat is moving at a true speed of 11 miles per hour (mi/hr) in a southerly direction. The current is known to be from the northeast at 3 mi/hr. What is the speed of the motorboat relative to the water? In what direction does the compass indicate that the boat is headed?

Computing Work Find the work done by a force of 5 pounds acting in the direction 60 to the horizontal in moving an object 20 feet from (0,0) to (20,0).