Now solved: In Exercises 6972, determine which of the vectors is (are) parallel to Use a

In Exercises 1 and 2, approximate the coordinates of the points.

In Exercises 1 and 2, approximate the coordinates of the points.

In Exercises 3-6, plot the points on the same three-dimensional coordinate system. (a) (2, 1,3) (b) (-1, 2, 1)

In Exercises 3-6, plot the points on the same three-dimensional coordinate system. (a) (3, -2, 5) (b) (\(\frac{3}{2}\), 4, -2) Text Transcription: 3/2

In Exercises 3-6, plot the points on the same three-dimensional coordinate system. (a) (5, -2, 2) (b) (5, -2, -2)

In Exercises 3-6, plot the points on the same three-dimensional coordinate system. (a) (0, 4, -5) (b) (4, 0, 5)

In Exercises 7-10, find the coordinates of the point. The point is located three units behind the yz-plane, four units to the right of the xz-plane, and five units above the xy-plane.

In Exercises 7-10, find the coordinates of the point. The point is located seven units in front of the yz-plane, two units to the left of the xz-plane, and one unit below the xy-plane.

In Exercises 7-10, find the coordinates of the point. The point is located on the x-axis, 12 units in front of the yz-plane.

In Exercises 7-10, find the coordinates of the point. The point is located in the yz-plane, three units to the right of the xz-plane, and two units above the xy-plane.

Think About It What is the z-coordinate of any point in the xy-plane?

Think About It What is the x-coordinate of any point in the yz-plane?

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(z=6\) Text Transcription: z=6

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(y=2\) Text Transcription: y=2

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(x=-3\) Text Transcription: x=-3

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(z=-\frac{5}{2}\) Text Transcription: z=-5/2

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(y<0\) Text Transcription: y<0

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(x>0\) Text Transcription: x>0

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(|y| \leq 3\) Text Transcription: |y| leq 3

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(|x|>4\) Text Transcription: |x|>4

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(x y>0, \quad z=-3\) Text Transcription: xy>0, z=-3

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(x y<0, \quad z=4\) Text Transcription: x y<0, z=4

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(x y z<0\) Text Transcription: xyz<0

In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). \(x y z>0\) Text Transcription: xyz>0

In Exercises 25-28, find the distance between the points. (0, 0, 0), (-4, 2, 7)

In Exercises 25-28, find the distance between the points. (-2, 3, 2), (2, -5, -2)

In Exercises 25-28, find the distance between the points. (1, -2, 4), (6, -2, -2)

In Exercises 25-28, find the distance between the points. (2, 2, 3), (4, -5, 6)

In Exercises 29-32, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (0, 0, 4), (2, 6, 7), (6, 4, -8)

In Exercises 29-32, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (3, 4, 1), (0, 6, 2), (3, 5, 6)

In Exercises 29-32, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (-1, 0, -2). (-1, 5, 2). (-3, -1, 1)

In Exercises 29-32, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (4, -1, -1), (2, 0, -4), (3, 5, -1)

Think About lt The triangle in Exercise 29 is translated five units upward along the z-axis. Determine the coordinates of the translated triangle.

Think About It The triangle in Exercise 30 is translated three units to the right along the y-axis. Determine the coordinates of the translated triangle.

In Exercises 35 and 36, find the coordinates of the midpoint of the line segment joining the points. (5, -9, 7), (-2, 3, 3)

In Exercises 35 and 36, find the coordinates of the midpoint of the line segment joining the points. (4, 0, -6), (8, 8, 20)

In Exercises 37-40, find the standard equation of the sphere. Center: (0, 2, 5) Radius: 2

In Exercises 37-40, find the standard equation of the sphere. Center: (4, -1, 1) Radius: 5

In Exercises 37-40, find the standard equation of the sphere. Endpoints of a diameter: (2, 0, 0), (0, 6, 0)

In Exercises 37-40, find the standard equation of the sphere. Center: (-3, 2, 4), tangent to the yz-plane

In Exercises 41-44, complete the square to write the equation of the sphere in standard form. Find the center and radius. \(x^{2}+y^{2}+z^{2}-2 x+6 y+8 z+1=0\) Text Transcription: x^2+y^2+z^2-2 x+6 y+8 z+1=0

In Exercises 41-44, complete the square to write the equation of the sphere in standard form. Find the center and radius. \(x^{2}+y^{2}+z^{2}+9 x-2 y+10 z+19=0\) Text Transcription: x^2+y^2+z^2+9 x-2 y+10 z+19=0

In Exercises 41-44, complete the square to write the equation of the sphere in standard form. Find the center and radius. \(9 x^{2}+9 y^{2}+9 z^{2}-6 x+18 y+1=0\) Text Transcription: 9 x^2+9 y^2+9 z^2-6 x+18 y+1=0

In Exercises 41-44, complete the square to write the equation of the sphere in standard form. Find the center and radius. \(4 x^{2}+4 y^{2}+4 z^{2}-24 x-4 y+8 z-23=0\) Text Transcription: 4 x^2+4 y^2+4 z^2-24 x-4 y+8 z-23=0

In Exercises 45-48, describe the solid satisfying the condition. \(x^{2}+y^{2}+z^{2} \leq 36\) Text Transcription: x^2+y^2+z^2 leq 36

In Exercises 45-48, describe the solid satisfying the condition. \(x^{2}+y^{2}+z^{2}>4\) Text Transcription: x^2+y^2+z^2 > 4

In Exercises 45-48, describe the solid satisfying the condition. \(x^{2}+y^{2}+z^{2}<4 x-6 y+8 z-13\) Text Transcription: x^2+y^2+z^2<4 x-6 y+8 z-13

In Exercises 45-48, describe the solid satisfying the condition. \(x^{2}+y^{2}+z^{2}>-4 x+6 y-8 z-13\) Text Transcription: x^2+y^2+z^2>-4 x+6 y-8 z-13

In Exercises 49–52, (a) find the component form of the vector v, (b) write the vector using standard unit vector notation, and (c) sketch the vector with its initial point at the origin.

In Exercises 49–52, (a) find the component form of the vector v, (b) write the vector using standard unit vector notation, and (c) sketch the vector with its initial point at the origin.

In Exercises 49–52, (a) find the component form of the vector v, (b) write the vector using standard unit vector notation, and (c) sketch the vector with its initial point at the origin.

In Exercises 49–52, (a) find the component form of the vector v, (b) write the vector using standard unit vector notation, and (c) sketch the vector with its initial point at the origin.

In Exercises 53-56, find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial Point Terminal Point (3, 2, 0) (4, 1, 6)

In Exercises 53-56, find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial Point Terminal Point (4, -5, 2) (-1, 7, -3)

In Exercises 53-56, find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial Point Terminal Point (-4, 3, 1) (-5, 3, 0)

In Exercises 53-56, find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial Point Terminal Point (1, -2, 4) (2, 4, -2)

In Exercises 57 and 58, the initial and terminal points of a vector v are given. (a) Sketch the directed line segment, (b) find the component form of the vector, (c) write the vector using standard unit vector notation, and (d) sketch the vector with its initial point at the origin. Initial point: (-1, 2, 3) Terminal point: (3, 3, 4)

In Exercises 57 and 58, the initial and terminal points of a vector v are given. (a) Sketch the directed line segment, (b) find the component form of the vector, (c) write the vector using standard unit vector notation, and (d) sketch the vector with its initial point at the origin. Initial point: (2, -1, -2) Terminal point: (-4, 3, 7)

In Exercises 59 and 60, the vector v and its initial point are given. Find the terminal point. \(\mathbf{v}=\langle 3,-5,6\rangle\) Initial point: (0, 6, 2) Text Transcription: v = <3, -5, 6>

In Exercises 59 and 60, the vector v and its initial point are given. Find the terminal point. \(\mathbf{v}=\left\langle 1,-\frac{2}{3}, \frac{1}{2}\right\rangle\) Initial point: (0, 2, \(\frac{5}{2}\)) Text Transcription: v = <1, -2/3, 1/2> 5/2

In Exercises 61 and 62, find each scalar multiple of v and sketch its graph. \(\mathbf{v}=\langle 1,2,2\rangle\) (a) 2v (b) -v (c) \(\frac{3}{2} \mathbf{v}\) (d) 0v Text Transcription: v=<1,2,2> 3/2v

In Exercises 61 and 62, find each scalar multiple of v and sketch its graph. \(\mathbf{v}=\langle 2,-2,1\rangle\) (a) -v (b) 2v (c) \(\frac{1}{2} \mathbf{v}\) (d) \(\frac{5}{2} \mathbf{v}\) Text Transcription: v=<2,-2,1> 1/2v 5/2v

In Exercises 63-68, find the vector z, given that \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}, 3\rangle\), \(\mathbf{v}=\langle\mathbf{2}, \mathbf{2},-\mathbf{1}\rangle\), and \(\mathbf{w}=\langle\boldsymbol{4}, \mathbf{0},-\mathbf{4}\rangle\). z = u - v Text Transcription: u=<1, 2, 3> v=<2, 2, -1> w=<4, 0, -4>

In Exercises 63-68, find the vector z, given that \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}, 3\rangle\), \(\mathbf{v}=\langle\mathbf{2}, \mathbf{2},-\mathbf{1}\rangle\), and \(\mathbf{w}=\langle\boldsymbol{4}, \mathbf{0},-\mathbf{4}\rangle\). z = u - v + 2w Text Transcription: u=<1, 2, 3> v=<2, 2, -1> w=<4, 0, -4>

In Exercises 63-68, find the vector z, given that \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}, 3\rangle\), \(\mathbf{v}=\langle\mathbf{2}, \mathbf{2},-\mathbf{1}\rangle\), and \(\mathbf{w}=\langle\boldsymbol{4}, \mathbf{0},-\mathbf{4}\rangle\). z = 2u + 4v - w Text Transcription: u=<1, 2, 3> v=<2, 2, -1> w=<4, 0, -4>

In Exercises 63-68, find the vector z, given that \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}, 3\rangle\), \(\mathbf{v}=\langle\mathbf{2}, \mathbf{2},-\mathbf{1}\rangle\), and \(\mathbf{w}=\langle\boldsymbol{4}, \mathbf{0},-\mathbf{4}\rangle\). \(\mathbf{z}=5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} \mathbf{w}\) Text Transcription: u=<1, 2, 3> v=<2, 2, -1> w=<4, 0, -4> z = 5u-3v-1/2w

In Exercises 63-68, find the vector z, given that \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}, 3\rangle\), \(\mathbf{v}=\langle\mathbf{2}, \mathbf{2},-\mathbf{1}\rangle\), and \(\mathbf{w}=\langle\boldsymbol{4}, \mathbf{0},-\mathbf{4}\rangle\). 2z - 3u = w Text Transcription: u=<1, 2, 3> v=<2, 2, -1> w=<4, 0, -4>

In Exercises 63-68, find the vector z, given that \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}, 3\rangle\), \(\mathbf{v}=\langle\mathbf{2}, \mathbf{2},-\mathbf{1}\rangle\), and \(\mathbf{w}=\langle\boldsymbol{4}, \mathbf{0},-\mathbf{4}\rangle\). 2u + v w + 3z = 0 Text Transcription: u=<1, 2, 3> v=<2, 2, -1> w=<4, 0, -4>

In Exercises 69-72, determine which of the vectors is (are) parallel to z. Use a graphing utility to confirm your results. \(\mathbf{z}=\langle 3,2,-5\rangle\) (a) \(\langle-6,-4,10\rangle\) (b) \(\left\langle 2, \frac{4}{3},-\frac{10}{3}\right\rangle\) (c) \(\langle 6,4,10\rangle\) (d) \(\langle 1,-4,2\rangle\) Text Transcription: z=<3,2,-5> <-6,-4,10> <2,4/3,-10/3> <6,4,10> <1,-4,2>

In Exercises 69-72, determine which of the vectors is (are) parallel to z. Use a graphing utility to confirm your results. \(\mathbf{z}=\frac{1}{2} \mathbf{i}-\frac{2}{3} \mathbf{j}+\frac{3}{4} \mathbf{k}\) (a) \(6 \mathbf{i}-4 \mathbf{j}+9 \mathbf{k}\) (b) \(-\mathbf{i}+\frac{4}{3} \mathbf{j}-\frac{3}{2} \mathbf{k}\) (c) \(12 \mathbf{i}+9 \mathbf{k}\) (d) \(\frac{3}{4} \mathbf{i}-\mathbf{j}+\frac{9}{8} \mathbf{k}\) Text Transcription: z=1/2i-2/3j+3/4k 6i-4j+9k -i+4/3j-3/2k 12i+9k 3/4i-j+9/8k

z has initial point (1,-1, 3) and terminal point (-2, 3, 5). (a) -6i + 8j + 4k (b) 4j + 2k

z has initial point (5, 4, 1) and terminal point (-2, -4, 4). (a) \(\langle 7,6,2\rangle\) (b) \(\langle 14,16,-6\rangle\) Text Transcription: <7, 6, 2> <14, 16, -6>

In Exercises 73-76, use vectors to determine whether the points are collinear. (0, -2, -5), (3, 4, 4), (2, 2, 1)

In Exercises 73-76, use vectors to determine whether the points are collinear. (4, -2, 7), (-2, 0, 3), (7, -3, 9)

In Exercises 73-76, use vectors to determine whether the points are collinear. (1,2, 4), (2, 5, 0), (0, 1, 5)

In Exercises 73-76, use vectors to determine whether the points are collinear. (0, 0, 0), (1, 3, -2), (2, -6, 4)

In Exercises 77 and 78, use vectors to show that the points form the vertices of a parallelogram. (2, 9, 1), (3, 11, 4), (0, 10, 2), (1, 12, 5)

In Exercises 77 and 78, use vectors to show that the points form the vertices of a parallelogram. (1, 1, 3), (9, -1, -2). (11, 2, -9). (3, 4, -4)

In Exercises 79-84, find the magnitude of v. \(\mathbf{v}=\langle 0,0,0\rangle\) Text Transcription: v=<0, 0, 0>

In Exercises 79-84, find the magnitude of v. \(\mathbf{v}=\langle 1,0,3\rangle\) Text Transcription: v=<1, 0, 3>

In Exercises 79-84, find the magnitude of v. v = 3j - 5k

In Exercises 79-84, find the magnitude of v. v = 2i + 5j - k

In Exercises 79-84, find the magnitude of v. v = i - 2j - 3k

In Exercises 79-84, find the magnitude of v. v = - 4i + 3j + 7k

In Exercises 85-88, find a unit vector (a) in the direction of v and (b) in the direction opposite of v. \(\mathbf{v}=\langle 2,-1,2\rangle\) Text Transcription: v=<2, -1, 2>

In Exercises 85-88, find a unit vector (a) in the direction of v and (b) in the direction opposite of v. \(\mathbf{v}=\langle 6,0,8\rangle\) Text Transcription: v=<6, 0, 8>

In Exercises 85-88, find a unit vector (a) in the direction of v and (b) in the direction opposite of v. \(\mathbf{v}=\langle 3,2,-5\rangle\) Text Transcription: v=<3, 2, -5>

In Exercises 85-88, find a unit vector (a) in the direction of v and (b) in the direction opposite of v. \(\mathbf{v}=\langle 8,0,0\rangle\) Text Transcription: v=<8, 0, 0>

Programming You are given the component forms of the vectors u and v. Write a program for a graphing utility in which the output is (a) the component form of u + v, (b) ||u + v|| (c) ||u||, and (d) ||v||. (e) Run the program for the vectors \(\mathbf{u}=\langle-1,3,4\rangle\) and \(\mathbf{v}=\langle 5,4.5,-6\rangle\). Text Transcription: u=<-1, 3, 4> v=<5, 4.5, -6>

Consider the two nonzero vectors u and v, and let s and t be real numbers. Describe the geometric figure generated by the terminal points of the three vectors tv, u + tv, and su + tv.

In Exercises 91 and 92, determine the values of c that satisfy the equation. Let u = -i + 2j + 3k and v = 2i + 2j - k. ||cv|| = 7

In Exercises 91 and 92, determine the values of c that satisfy the equation. Let u = -i + 2j + 3k and v = 2i + 2j - k. ||cu|| = 4

In Exercises 93-96, find the vector v with the given magnitude and direction u. Magnitude Direction 10 \(\mathbf{u}=\langle 0,3,3\rangle\) Text Transcription: u=<0,3,3>

In Exercises 93-96, find the vector v with the given magnitude and direction u. Magnitude Direction 3 \(\mathbf{u}=\langle 1,1,1\rangle\) Text Transcription: u=<1,1,1>

In Exercises 93-96, find the vector v with the given magnitude and direction u. Magnitude Direction \(\frac{3}{2}\) \(\mathbf{u}=\langle 2,-2,1\rangle\) Text Transcription: 3/2 u=<2,-2,1>

In Exercises 93-96, find the vector v with the given magnitude and direction u. Magnitude Direction 7 \(\mathbf{u}=\langle -4,6,2\rangle\) Text Transcription: u=<-4,6,2>

In Exercises 97 and 98, sketch the vector v and write its component form. v lies in the yz-plane, has magnitude 2, and makes an angle of \(30^{\circ}\) with the positive y-axis. Text Transcription: 30 degrees

In Exercises 97 and 98, sketch the vector v and write its component form. v lies in the xz-plane, has magnitude 5, and makes an angle of \(45^{\circ}\) with the positive z-axis. Text Transcription: 45 degrees

In Exercises 99 and 100, use vectors to find the point that lies two-thirds of the way from P to Q. P(4, 3, 0), Q(1, -3, 3)

In Exercises 99 and 100, use vectors to find the point that lies two-thirds of the way from P to Q. P(1, 2, 5), Q6. 8, 2)

Let u = i +j, v = j + k, and w = au + bv. (a) Sketch u and v. (b) If w = 0, show that a and b must both be zero. (c) Find a and b such that w = i + 2j + k. (d) Show that no choice of a and b yields w = i + 2j + 3k.

Writing The initial and terminal points of the vector v are \(\left(x_{1}, y_{1}, z_{1}\right)\) and (x, y, z). Describe the set of all points (x, y, z) such that ||v|| = 4. Text Transcription: (x_1, y_1, z_1)

A point in the three-dimensional coordinate system has coordinates \(\left(x_{0}, y_{0}, z_{0}\right)\). Describe what each coordinate measures. Text Transcription: (x_0, y_0, z_0)

Give the formula for the distance between the points \(\left(x_{1}, y_{1}, z_{1}\right)\) and \(\left(x_{2}, y_{2}, z_{2}\right)\). Text Transcription: (x_1, y_1, z_1) (x_2, y_2, z_2)

Give the standard equation of a sphere of radius r, centered at \(\left(x_{0}, y_{0}, z_{0}\right)\). Text Transcription: (x_0, y_0, z_0)

State the definition of parallel vectors.

Let A, B, and C be vertices of a triangle. Find \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}\). Text Transcription: over rigtharrow AB + over rigtharrow BC + over rigtharrow CA

Let \(\mathbf{r}=\langle x, y, z\rangle\) and \(\mathbf{r}_{0}=\langle 1,1,1\rangle\). Describe the set of all points (x, y, z) such that \(\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=2\). Text Transcription: r=<x,y,z> r_0=<1,1,1> ||r-r_0||=2

Numerical, Graphical, and Analytic Analysis The lights in an auditorium are 24-pound discs of radius 18 inches. Each disc is supported by three equally spaced cables that are L inches long (see figure). (a) Write the tension T in each cable as a function of L. Determine the domain of the function. (b) Use a graphing utility and the function in part (a) to complete the table. (c) Use a graphing utility to graph the function in part (a). Determine the asymptotes of the graph. (d) Confirm the asymptotes of the graph in part (c) analytically. (e) Determine the minimum length of each cable if a cable is designed to carry a maximum load of 10 pounds.

Think About It Suppose the length of each cable in Exercise 109 has a fixed length L = a, and the radius of each dise is \(r_{0}\) inches. Make a conjecture about the limit \(\lim _{r_{0} \rightarrow a^{-}}\) T and give a reason for your answer. Text Transcription: r_0 lim_r_0 rightarrow a^-

Diagonal of a Cube Find the component form of the unit vector v in the direction of the diagonal of the cube shown in the figure.

Tower Guy Wire The guy wire supporting a 100-foot tower has a tension of 550 pounds. Using the distances shown in the figure, write the component form of the vector F representing the tension in the wire.

Load Supports Find the tension in each of the supporting cables in the figure if the weight of the crate is 500 newtons.

Construction A precast concrete wall is temporarily kept in its vertical position by ropes (see figure). Find the total force exerted on the pin at position A. The tensions in AB and AC are 420 pounds and 650 pounds.

Write an equation whose graph consists of the set of points P(x, y, z) that are twice as far from A(0, -1, 1) as from B(1, 2, 0).