Solved: Use the definition of scalar product, a b = ab cos

Can the sum of the magnitudes of two vectors ever be equal to the magnitude of the sum of the same two vectors? If no, why not? If yes, when?

The two vectors shown in Fig. 3-21 lie in an xy plane. What are the signs of the x and y components, respectively, of (a) ~ + Ch, (b) ~ - Ch,and (c) Ch - ~?

Being part of the "Gators," the University of Florida golfing team must play on a putting green with an alligator pit. Figure 3-22 shows an overhead view of one putting challenge of the team; an xy coordinate system is superimposed. Team members must putt from the origin to the hole, which is at xy coordinates (8 m, 12 m), but they can putt the golf ball using only one or more of the following displacements, one or more times: -a: = (8 m)i + (6 m)], Ch = (6 m)], C4 = (8 m)i. The pit is at coordinates (8 m, 6 m). If a team member putts the ball into or through the pit, the member is automatically transferred to Florida State University, the arch rival. What sequence of displacements should a team member use to avoid the pit?

Equation 3-2 shows that the addition of two vectors a and b is commutative. Does that mean subtraction is commutative, so that a - b = b -a?

Which of the arrangements of axes in Fig. 3-23 can be labeled "right-handed coordinate system"? As usual, each axis label indicates the positive side of the axis.

escribe two vectors a and b such that (a) a + b = e and a + b = c; (b) a + b = a - b; (c) a + b = e and a2 + b2 = c2

If d = a + b + (-e), does (a) a + (-d) = e + (-b), (b) a = (-b) + d + e,and (c)e + (-d) = a + b?

Ifab = ae,mustb equale?

If F = q(\1 x E) and \1 is perpendicular to 13, then what is the direction of 13 in the three situations shown in Fig. 3-24 when constant q is (a) positive and (b) negative?

Figure 3-25 shows vector A and four other vectors that have the same magnitude but differ in orientation. (a) Which of those other four vectors have the same dot product with A? (b) Which have a negative dot product with A?

(a) In unit-vector notation, what is the sum a + b if a = (4.0 m)i + (3.0 m)] and b = (-13.0 m)i + (7.0 m)]? What are the (b) magnitude and (c) direction ofa + b?

A car is driven east for a distance of 50 km, then north for 30 km, and then in a direction 30 east of north for 25 km. Sketch the vector diagram and determine (a) the magnitude and (b) the angle of the car's total displacement from its starting point.

A person desires to reach a point that is 3.40 km from her present location and in a direction that is 35.0 north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

You are to make four straight-line moves over a flat desert floor, starting at the origin of an xy coordinate system and ending at the xy coordinates (-140 m, 30 m). The x component and y component of your moves are the following, respectively, in meters: (20 and 60), then (b, and -70), then (-20 and c y ), then (-60 and -70). What are (a) component b, and (b) component c,,? What are (c) the magnitude and (d) the angle (relative to the positive direction of the x axis) of the overall displacement?

The two vectors a and b in Fig. 3-28 have equal magnitudes of 10.0 m and the angles are 81 = 30 and 82 = 105. Find the (a) x and (b) Y com- Y ponents of their vector sum r, (c) the magnitude of 1, and (d) the angle 1 makes with the positive direction of the x axis.

For the displacement vectors a = (3.0 m)i + (4.0 m)J and b = (5.0 m)i + (-2.0 m)], give a + bin (a) unit-vector notation, and as (b) a ""---'-_______ x magnitude and (c) an angle (rela- 0 tive to i). Now give b - a in (d) Fig.3-28 Problem 15. unit-vector notation, and as (e) a magnitude and (f) an angle.

Three vectors a, b, and c each have a magnitude of 50 m and lie in an xy plane. Their directions relative to the positive direction of the x axis are 30, 195, and 315, respectively. What are (a) the magnitude and (b) the angle of the vector 7i + b + c, and (c) the magnitude and (d) the angle of a - b + c? What are the (e) magnitude and (f) angle of a fourth vector d such that (a + b) - (c + d) = O?

In the sum A + B = C, vector A has a magnitude of 12.0 m and is angled 40.0 counterclockwise from the +x direction, and vector C has a magnitude of 15.0 m and is angled 20.0 counterclockwise from the -x direction. What are (a) the magnitude and (b) the angle (relative to +x) ofB?

In a game of lawn chess, where pieces are moved between the centers of squares that are each 1.00 m on edge, a knight is moved in the following way: (1) two squares forward, one square rightward; (2) two squares leftward, one square forward; (3) two squares forward, one square leftward. What are (a) the magnitude and (b) the angle (relative to "forward") of the knight's overall displacement for the series of three moves?

An explorer is caught in a white out (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for 5.6 km, but when the snow clears, he discovers that he actually traveled 7.8 km at 50 north of due east. (a) How far and (b) in what direction must he now travel to reach base camp?

An ant, crazed by the Sun on a hot Texas afternoon, darts over an xy plane scratched in the dirt. The x and y components of four consecutive darts are the following, all in centimeters: (30.0, 40.0), (bt , -70.0), (-20.0, cy), (-80.0, -70.0). The overall displacement of the four darts has the xy components (-140, -20.0). What are (a) bt and (b) cy? What are the ( c) magnitude and (d) angle (relative to the positive direction of the x axis) of the overall displacement?

(a) What is the sum of the following four vectors in unitvector notation? For that sum, what are (b) the magnitude, (c) the angle in degrees, and (d) the angle in radians? If: 6.00 m at +0.900 rad G: 4.00 m at + 1.20 rad F: 5.00 mat -75.0 H: 6.00 mat -210

If B is added to C = 3.01 + 4.0J, the result is a vector in the positive direction of the y axis, with a magnitude equal to that of C. What is the magnitude of B?

Vector A, which is directed along an x axis, is to be added to vector B, which has a magnitude of 7.0 m. The sum is a third vector that is directed along the y axis, with a magnitude that is 3.0 times that of A. What is that magnitude of A?

Oasis B is 25 km due east of oasis A. Starting from oasis A, a camel walks 24 km in a direction 15 south of east and then walks 8.0 km due north. How far is the camel then from oasis B?

What is the sum of the following four vectors in (a) unit-vector notation, and as (b) a magnitude and ( c) an angle? A = (2.00 m)i + (3.00 m)J B: 4.00 m, at +65.0 C = (-4.00 m)i + (-6.00 m)J 13: 5.00 m, at -235

If dl + d2 = 5d3, dl - d2 = 3d3, and d3 = 2i + 4], then what are, in unit-vector notation, (a) dl and (b) d2?

Two beetles run across fiat sand, starting at the same point. Beetle 1 runs 0.50 m due east, then 0.80 m at 30 north of due east. Beetle 2 also makes two runs; the first is 1.6 m at 40 east of due north. What must be (a) the magnitude and (b) the direction of its second run if it is to end up at the new location of beetle I?

Typical backyard ants often create a network of chemical trails for guidance. Extending outward from the nest, a trail branches (bifurcates) repeatedly, with 60 between the branches. If a roaming ant chances upon a trail, it can tell the way to the nest at any branch point: If it is moving away from the nest, it has two choices of path requiring a small turn in its travel direction, either 30 leftward or 30 rightward. If it is moving toward the nest, it has only one such choice. Figure 3-29 shows a typical ant trail, with lettered straight sections of 2.0 cm length and symmetric bifurcation of 60. Path v is parallel to the y axis. What are the (a) magnitUde and (b) angle (relative to the positive direction of the superimposed x axis) of an ant's displacement from the nest (find it in the figure) if the ant enters the trail at point A? What are the (c) magnitUde and (d) angle if it enters at point B?

Here are two vectors: L, III 10 a = (4.0 m)1 - (3.0 m)J and b = (6.0 m)i + (8.0 m)]. What are (a) the magnitude and (b) the angle (relative to i) of 7i? What are (c) the magnitUde and (d) the angle ofb? What are (e) the magnitude and (f) the angle of a + b; z ~) the magnitude and (h) the angle of b a; and (i) the magnitUde and (j) the angle ofa - b? (k) What is the angle between the directions of b - a anda-b?

In Fig. 3-30, a cube of edge length a sits with one corner at the ori- gin of an xyz coordinate system. A body diagonal is a line that extends from one corner to another through the center. In unit-vector notation, what is the body diagonal that extends from the corner at (a) coordinates (0, 0, 0), (b) coordinates (a, 0, 0), (c) coordinates (0, a, 0), and (d) coordinates (a, a, O)? (e) Determine the angles that the body diagonals make with the adjacent edges. (f) Determine the length of the body diagonals in terms of a.

Vectors and the Laws of Physics In Fig. 3-31, a vector a with a magnitude of 17.0 m is directed at angle e = 56.00 counterclockwise from the +x axis. What are the components (a) ax and (b) ay of the vector? A second coordinate system is inclined by angle e' = 18.00 with respect to the first. What are the components (c) a.~ and (d) a; in this primed coordinate system?

Multiplying Vectors For the vectors in Fig. 3-32, with a = 4, b = 3, and c = 5, what are (a) the magnitUde and (b) the direction of a x b, (c) the magnitude and (d) the direction of a x c, and (e) the magnitude and (f) the direction of b x c? (The z axis is not shown.)

Two vectors are presented as Fig. 3-32 Problems 33 and 54. a = 3.01 + 5.0] and b = 2.01 + 4.0. Find (a) a x b, (b) a' b, (c) (a + b) b, and (d) the component of a along the direction of b. (Hint: For (d), consider Eq. 3-20 and Fig. 3-18.)

Two vectors, rand s, lie in the xy plane. Their magnitudes are 4.50 and 7.30 units, respectively, and their directions are 3200 and 85.00 , respectively, as measured counterclockwise from the positive x axis. What are the values of (a) r sand (b) r x s?

If dl = 31 - 2J + 4k and d2 = -sf + 2] - k, then what is (dl + ( 2)' (d1 x 4(2)?

Three vectors are given by a = 3.01 + 3.0J - 2.0k, b = -1.01 - 4.0J + 2.0k, and c = 2.01 + 2.0J + 1.0k. Find (a) a'(b x c),(b)a(b + c), and (c)a x (b + c).

For the following three vectors, what is 3C' (2A x B)? A = 2.001 + 3.00J - 4.00k 13 = - 3.001 + 4.00J + 2.00k C = 7.001 - 8.00J

Vector A has a m~n1ude of 6.00 units, vector 13 has a magnitude of 7.00 units, and A B has a value of 14.0. What is the angle between the directions of A and 13?

Displacement d1 is in the yz plane 63.00 from the positive direction of the y axis, has a positive z component, and has a magnitude of 4.50 m. Displacement d2 is in the xz plane 30.00 from the positive direction of the x axis, has a positive z component, and has magnitude 1.40 m. What are (a) d1 d2, (b) dl x d2, and (c) the angle between d 1 and d 2?

Use the definition of scalar product, a b = ab cos e, and the fact that a b = arb, + ayby + alb, to calculate the angle between the two vectors given by a = 3.01 + 3.0] + 3.0k and b = 2.01 + 1.0J + 3.0k.

In a meeting of mimes, mime 1 goes through a displacement d1 = (4.0 m)l + (5.0 m)J and mime 2 goes through a displacement d.:2; = (.:3.0-->m)1 + (4.0 m)]. What are (aL d1 x d2, (b) d1 d2, (c) (d1 + d2) db and (d) the component of dl along the direction of d2? (Hint: For (d), see Eq. 3-20 and Fig. 3-18.)

The three vectors in Fig. 3-33 have magnitudes a =3.00 m, b = 4.00 m, and c = 10.0 m and angle e = 30.00 . What are (a) the x component and (b) the y component of a; (c) the x component and (d) the y component of b; and (e) the x component and (f) the y component of c? If c = pa + qb, what are the values of (g) p and (h) q?

In the product F = qv x 13, take q = 2, v = 2.01 + 4.0J + 6.0k and F = 4.01 - 20J + 12k. What then is 13 in unit-vector notation if B, = By?

Vectors A and 13 lie in an xy plane. A has magnitude 8.00 and angle 1300; 13 has components Br = -7.72 and B, = -9.20. (a) What is 5] . 13? What is 4A x 313 in (b) unit-vector notation and x z ~-------+----r--;--y / / / / (c) magnitude-angle notation . with spherical coordinates (see Fig. 3-34 Problem 45. Fig. 3-34)? (d) What is the an- --> --> gle between the directions of A and 4A x 3B? (Hint: Think a bit before you resort to a calculation.) What is A + 3.00k in (e) unitvector notation and (f) magnitude-angle notation with spherical coordinates?

Vector a has a magnitude of 5.0 m and is directed east. Vector b has a magnitude of 4.0 m and is directed 350 west of due north. What are (a) the magnitude and (b) the direction ofa + b? What are (c) the magnitude and (d) the direction of b - a? (e) Draw a vector diagram for each combination.

Vectors A and 13 lie in an xy plane. A has magnitude 8.00 and angle 1300; 13 has components B, = -7.72 and By = -9.20. What are the angles between the negative direction of the y axis and (a) the direction of A, (b) the direction of the product A x 13, and (c) the direction of A x (13 + 3.00k)?

Two vectors a and b have the components, in meters, a, = 3.2, ay = 1.6, b,. = 010, by = 4.5. (a) Find the angle between the directions of a and b. There are two vectors in the xy plane that are perpendicular to a and have a magnitude of 5.0 m. One, vector e, has a positive x component and the other, vector d, a negative x component. What are (b) the x component and (c) the y component of vector e, and (d) the x component and (e) the y component of vector d?

A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side, 90.0 km due north. The sailor, however, ends up 50.0 km due east of the starting point. (a) How far and (b) in what direction must the sailor now sail to reach the original destination?

Vector d1 is in the negative direction of a y axis, and vector d2 is in the positive direction of an x axis. What are the directions of 0) ~2/4 and (bl d11,-4)? What .are the.ma~nitudes of products (c) d1 d2 and (d) d 1 (d2 /4)? What IS the dlfectlOn of the vector resulting from (e) d1 X d2 and (f) d2 X d1? What is the magnitude of the vector product in (g) part (e) and (h) part (f)? What are the (i) magnitude and U) direction ofd1 X (d2/4)?

Rock faults are ruptures along which opposite faces of rock have slid past each other. In Fig. 3-35, points A and B coincided before the ro~ the foreground slid down to the right. The net displacemeni4B is along the plane of the fault. The horizo~ component of AB is the strike-slip AC. The component of AB that is directed down the plane of the fault ~he dip-slip AD. (a) What is the magnitude of the net displacement AB if the strike-slip is 22.0 m and the dip-slip is 17.0 m? (b) If the plane of the fault is inclined at ~e cjJ = 52.0 to the horizontal, what is the vertical component of AB ?

Here are three displacements, each measured in meters: = 4.01 + 5.0J - 6.0k, d2 = -1.01 + 2.0J + 3.0k, and d3 = 4.01 + 3.0J + 2.0k. (a) What is r = d1 - d2 + d3? (b) What is the angle between r and the positive z axis? (c) What is the component ofd! along the direction ofd2? (d) What is the component of d j that is perpendicular to the direction of d2 and in the plane of d! and d2? (Hint: For (c), consider Eq. 3-20 and Fig. 3-18; for (d), consider Eq. 3-27.)

A vector a of magnitude 10 units and another vector b of magnitude 6.0 units differ in directions by 60. Find (a) the scalar product of the two vectors and (b) the magnitude of the vector product a x b.

For the vectors in Fig. 3-32, with a = 4, b = 3, and c = 5, calculate (a) a b,(b) a' e,and (c) b e.

A particle undergoes three successive displacements in a plane, as follows: db 4.00 m southwest; then d2, 5.00 m east; and finally d3, 6.00 m in a direction 60.0 north of east. Choose a coordinate system with the y axis pointing north and the x axis pointing east. What are (a) the x component and (b) the y component of d1? What are (c) the x component and (d) the y component of PROBLEMS 57 d2? What are (e) the x component and (f) the y component of d3? Next, consider the net displacement of the particle for the three successive displacements. What are (g) the x component, (h) the y component, (i) the magnitude, and (j) the direction of the net displacement? If the particle is to return directly to the starting point, (k) how far and (I) in what direction should it move?

Find the sum of the following four vectors in (a) unit-vector notation, and as (b) a magnitude and (c) an angle relative to + x. P: 10.0 m, at 25.0 counterclockwise from +x Q: 12.0 m, at 10.0 counterclockwise from +y R: 8.00 m, at 20.0 clockwise from -y S: 9.00 m, at 40.0 counterclockwise from -y

If B is added to.II, the result is 6.01 + 1.0]. If B is subtracted from.II, the result is -4.01 + 7.0]. What is the magnitude of .II?

A vector d has a magnitude of 2.5 m and points north. What are (a) the magnitude and (b) the direction of 4.0d? What are (c) the magnitude and (d) the direction of - 3.0d?

I has the magnitude 12.0 m and is angled 60.0 counterclockwise from the positive direction of the x axis of an xy coordinate system. Also, B = (12.0 m)1 + (8.00 m)J on that same coordinate system. We now rotate the system counterclockwise about the origin by 20.0 to form anx'y' system. On this new system, what are (a).II and (b) B, both in unit-vector notation?

If a - b = 2e, a + b = 4e, and e = 31 + 4J, then what are (a) a and (b) b?

(a) In unit-vector notation, what is r = a - b + e if a = 5.01 + 4.0] - 6.0k, b = -2.01 + 2.0] + 3.01<:, and e = 4.01 + 3.0] + 2.0k? (b) Calculate the angle between r and the positive z axis. (c) What is the component ofa along the direction ofb? (d) What is the component of a perpendicular to the direction of b but in the plane of a and b? (Hint: For (c), see Eq. 3-20 and Fig. 3-18; for (d), see Eq. 3-27.)

A golfer takes three putts to get the ball into the hole. The first putt displaces the ball 3.66 m north, the second 1.83 m southeast, and the third 0.91 m southwest. What are (a) the magnitude and (b) the direction of the displacement needed to get the ball into the hole on the first putt?

Here are three vectors in meters: d1 = -3.01 + 3.0J + 2.0k d2 = -2.01 - 4.0] + 2.0k d3 = 2.01 + 3.0] + loOk. What results from (a) d1 (d2 + ( 3), (b) d j ' (d2 x ( 3), and (c) d1 X (d2 + ( 3)?

Consider two displacements, one of magnitude 3 m and another of magnitude 4 m. Show how the displacement vectors may be combined to get a resultant displacement of magnitude (a) 7 m, (b) 1 m, and ( c) 5 m.

A protester carries his sign of protest, starting from the origin of an xyz coordinate system, with the xy plane horizontal. He moves 40 m in the negative direction of the x axis, then 20 m along a perpendicular path to his left, and then 25 m up a water tower. (a) In unit-vector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?