Question 1

Write one or more inequalities that describe the set of points.
-The half-space consisting of the points on and behind the yz-plane

Accepted Answer

A)  $\mathrm { y } \leq 0 , \mathrm { z } \leq 0$ 
B)  $x \leq 0$ 
C)  $x > 0$ 
D)  $y > 0 , z > 0$ 
A)  $\mathrm { y } \leq 0 , \mathrm { z } \leq 0$ 
B)  $x \leq 0$ 
C)  $x > 0$ 
D)  $y > 0 , z > 0$

Question 2

Solve the problem.
-A bird flies from its nest $7 \mathrm {~km}$ in the direction $2 ^ { \circ }$ north of east, where it stops to rest on a tree. It then flies $10 \mathrm {~km}$ in the direction $4 ^ { \circ }$ south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the $x$-axis points east, and the $y$-axis points north, at what point is the telephone pole located?

Accepted Answer

A)  (6.996, 0.2443) 
B)  (-9.976, -0.6976) 
C)  (-2.980, -0.4533) 
D)  (3.623, 10.53) 
A)  (6.996, 0.2443) 
B)  (-9.976, -0.6976) 
C)  (-2.980, -0.4533) 
D)  (3.623, 10.53)

Question 3

Provide an appropriate response.
-Show that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. [Hint: See the figure below. Show that $\left| \frac { a + b } { 2 } \right| = \left| \frac { a - b } { 2 } \right|$.]

Accepted Answer

Verify that @#LAT-DLM&.
Cancel the 2 's and square

Question 4

Write the equation for the plane.
-The plane through the point A(2, 10, 8) perpendicular to the vector from the origin to A.

Accepted Answer

A)  $2 x + 10 y + 8 z = \sqrt { 168 }$ 
B)  $2 x + 10 y + 8 z = 168$ 
C)  $2 x + 10 y + 8 z = - 168$ 
D)  $2 x + 10 y + 8 z = 20$ 
A)  $2 x + 10 y + 8 z = \sqrt { 168 }$ 
B)  $2 x + 10 y + 8 z = 168$ 
C)  $2 x + 10 y + 8 z = - 168$ 
D)  $2 x + 10 y + 8 z = 20$

Question 5

Find a parametrization for the line segment joining the points.
-(2, -7, -2), (0, -7, 4)

Accepted Answer

A)  $x = 2 t , y = - 7 t , z = - 6 t + 4,0 \leq t \leq 1$ 
B)  $x = - 2 t + 2 , y = - 7 t , z = 6 t - 2,0 \leq t \leq 1$ 
C)  $x = - 2 t + 2 , y = - 7 , z = 6 t - 2,0 \leq t \leq 1$ 
D)  $x = 2 t , y = - 7 , z = - 6 t + 4,0 \leq t \leq 1$ 
A)  $x = 2 t , y = - 7 t , z = - 6 t + 4,0 \leq t \leq 1$ 
B)  $x = - 2 t + 2 , y = - 7 t , z = 6 t - 2,0 \leq t \leq 1$ 
C)  $x = - 2 t + 2 , y = - 7 , z = 6 t - 2,0 \leq t \leq 1$ 
D)  $x = 2 t , y = - 7 , z = - 6 t + 4,0 \leq t \leq 1$

Question 6

Solve the problem.
-Find the area of the triangle determined by the points P(-3, 6, -4), Q(-7, , ), and R(-8, -2, ).

Accepted Answer

A)  $\sqrt { 15410 }$ 
B)  $\sqrt { 13170 }$ 
C)  $\frac { \sqrt { 15410 } } { 2 }$ 
D)  $\frac { \sqrt { 13170 } } { 2 }$ 
A)  $\sqrt { 15410 }$ 
B)  $\sqrt { 13170 }$ 
C)  $\frac { \sqrt { 15410 } } { 2 }$ 
D)  $\frac { \sqrt { 13170 } } { 2 }$

Question 7

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$
-$\mathrm { P } _ { 1 } ( - 4,6,5 )$ and $\mathrm { P } _ { 2 } ( - 1,2,10 )$

Accepted Answer

A)  $\frac { 3 } { 5 \sqrt { 2 } } \mathrm { i } + \frac { - 4 } { 5 \sqrt { 2 } } \mathrm { j } + \frac { 1 } { \sqrt { 2 } } \mathrm { k } ; \left( - \frac { 5 } { 2 } , 4 , \frac { 15 } { 2 } \right)$ 
B)  $\frac { 3 } { 5 \sqrt { 2 } } \mathbf { i } + \frac { - 4 } { 5 \sqrt { 2 } } \mathrm { j } + \frac { 1 } { \sqrt { 2 } } \mathbf { k } ; \left( - \frac { 1 } { 2 } , 1,5 \right)$ 
C)  $\frac { 3 } { 50 } \mathbf { i } - \frac { 2 } { 25 } \mathrm { j } + \frac { 1 } { 10 } \mathbf { k } ; \left( - \frac { 5 } { 2 } , 4 , \frac { 15 } { 2 } \right)$ 
D)  $\frac { 3 } { 50 } \mathbf { i } - \frac { 2 } { 25 } \mathbf { j } + \frac { 1 } { 10 } \mathbf { k } ; \left( - 2,3 , \frac { 5 } { 2 } \right)$ 
A)  $\frac { 3 } { 5 \sqrt { 2 } } \mathrm { i } + \frac { - 4 } { 5 \sqrt { 2 } } \mathrm { j } + \frac { 1 } { \sqrt { 2 } } \mathrm { k } ; \left( - \frac { 5 } { 2 } , 4 , \frac { 15 } { 2 } \right)$ 
B)  $\frac { 3 } { 5 \sqrt { 2 } } \mathbf { i } + \frac { - 4 } { 5 \sqrt { 2 } } \mathrm { j } + \frac { 1 } { \sqrt { 2 } } \mathbf { k } ; \left( - \frac { 1 } { 2 } , 1,5 \right)$ 
C)  $\frac { 3 } { 50 } \mathbf { i } - \frac { 2 } { 25 } \mathrm { j } + \frac { 1 } { 10 } \mathbf { k } ; \left( - \frac { 5 } { 2 } , 4 , \frac { 15 } { 2 } \right)$ 
D)  $\frac { 3 } { 50 } \mathbf { i } - \frac { 2 } { 25 } \mathbf { j } + \frac { 1 } { 10 } \mathbf { k } ; \left( - 2,3 , \frac { 5 } { 2 } \right)$

Question 8

Write the equation for the plane.
-The plane through the point P(-6, 8, 9) and perpendicular to the line x = 3 + 9t, y = -5 + 9t, z = 6 - t.

Accepted Answer

A)  9x + 9y + z = 9 
B)  9x + 9y - z = 9 
C)  9x + 9y - z = 27 
D)  9x + 9y - z = -9 
A)  9x + 9y + z = 9 
B)  9x + 9y - z = 9 
C)  9x + 9y - z = 27 
D)  9x + 9y - z = -9

Question 9

Identify the type of surface represented by the given equation.
-$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } - \frac { z ^ { 2 } } { 9 } = 1$$

Accepted Answer

A)  Elliptical cone 
B)  Ellipsoid 
C)  Hyperboloid of two sheets 
D)  Hyperboloid of one sheet 
A)  Elliptical cone 
B)  Ellipsoid 
C)  Hyperboloid of two sheets 
D)  Hyperboloid of one sheet

Question 10

Write the equation for the plane.
-The plane through the point P(-2, -5, 6) and normal to n = -6i - 3j + 5k.

Accepted Answer

A)  -2x - 5y - 6z = -33 
B)  6x + 3y - 5z = -33 
C)  -6x - 3y + 5z = 57 
D)  2x + 5y + 6z = -33 
A)  -2x - 5y - 6z = -33 
B)  6x + 3y - 5z = -33 
C)  -6x - 3y + 5z = 57 
D)  2x + 5y + 6z = -33

Question 11

Solve the problem.
-An airplane is flying in the direction $51 ^ { \circ }$ west of north at $660 \mathrm {~km} / \mathrm { hr }$. Find the component form of the velocity of the airplane, assuming that the positive $x$-axis represents due east and the positive $y$-axis represents due north.

Accepted Answer

A)  $\langle - 442.4,489.8 \rangle$ 
B)  $\langle - 0.7771,0.6293 \rangle$ 
C)  $\langle - 415.4,512.9 \rangle$ 
D)  $\langle - 512.9,415.4 \rangle$ 
A)  $\langle - 442.4,489.8 \rangle$ 
B)  $\langle - 0.7771,0.6293 \rangle$ 
C)  $\langle - 415.4,512.9 \rangle$ 
D)  $\langle - 512.9,415.4 \rangle$

Question 12

Find the center and radius of the sphere.
-$$x ^ { 2 } + ( y + 10 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 64$$

Accepted Answer

A)  C(0, 10, -1), a = 64 
B)  C(0, -10, 1), a = 64 
C)  C(0, 10, -1), a = 8 
D)  C(0, -10, 1), a = 8 
A)  C(0, 10, -1), a = 64 
B)  C(0, -10, 1), a = 64 
C)  C(0, 10, -1), a = 8 
D)  C(0, -10, 1), a = 8

Question 13

Provide an appropriate response.
-$\text { Show that the vectors } | \mathbf { a } | \mathbf { b } + | \mathbf { b } | \mathbf { a } \text { and } | \mathbf { a } | \mathbf { b } - | \mathbf { b } | \mathbf { a } \text { are orthogonal. }$

Accepted Answer

Take the dot product:
Since th

Question 14

Write one or more inequalities that describe the set of points.
-The exterior of the sphere of radius 5 centered at the point (-2, -3, -5)

Accepted Answer

A) $( x + 2 ) ^ { 2 } + ( x + 3 ) ^ { 2 } + ( x + 5 ) ^ { 2 } > 25$ B) $( x - 2 ) ^ { 2 } + ( x - 3 ) ^ { 2 } + ( x - 5 ) ^ { 2 } < 25$ C) $( x - 2 ) ^ { 2 } + ( x - 3 ) ^ { 2 } + ( x - 5 ) ^ { 2 } \geq 25$ D) $( x + 2 ) ^ { 2 } + ( x + 3 ) ^ { 2 } + ( x + 5 ) ^ { 2 } < 25$ A) $( x + 2 ) ^ { 2 } + ( x + 3 ) ^ { 2 } + ( x + 5 ) ^ { 2 } > 25$ B) $( x - 2 ) ^ { 2 } + ( x - 3 ) ^ { 2 } + ( x - 5 ) ^ { 2 } < 25$ C) $( x - 2 ) ^ { 2 } + ( x - 3 ) ^ { 2 } + ( x - 5 ) ^ { 2 } \geq 25$ D) $( x + 2 ) ^ { 2 } + ( x + 3 ) ^ { 2 } + ( x + 5 ) ^ { 2 } < 25$

Question 15

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (10, 11); perpendicular to v = 5i - 4j

Accepted Answer

A)  5x - 4y = 41 
B)  y - 11 = 3(x - 5) 
C)  -4x - 5y = -95 
D)  5x - 4y = 6 
A)  5x - 4y = 41 
B)  y - 11 = 3(x - 5) 
C)  -4x - 5y = -95 
D)  5x - 4y = 6

Question 16

Find the angle between u and v in radians.
-u = 7i + 6j + 3k, v = 5i + 2j + 6k

Accepted Answer

A)  1.15 
B)  0.98 
C)  0.59 
D)  1.47 
A)  1.15 
B)  0.98 
C)  0.59 
D)  1.47

Question 17

Find the intersection.
-x = 4 + 3t, y = 10 - 10t, z = 2 + 3t ; -2x + 2y - 2z = 9

Accepted Answer

A)  $\left( \frac { 125 } { 32 } , \frac { 165 } { 16 } , \frac { 61 } { 32 } \right)$ 
B)  $( 1,20 , - 1 )$ 
C)  $\left( \frac { 131 } { 32 } , \frac { 155 } { 16 } , \frac { 63 } { 32 } \right)$ 
D)  $( 7,0,5 )$ 
A)  $\left( \frac { 125 } { 32 } , \frac { 165 } { 16 } , \frac { 61 } { 32 } \right)$ 
B)  $( 1,20 , - 1 )$ 
C)  $\left( \frac { 131 } { 32 } , \frac { 155 } { 16 } , \frac { 63 } { 32 } \right)$ 
D)  $( 7,0,5 )$

Question 18

Find the triple scalar product (u x v) · w of the given vectors.
-u = 4i + 2j - k; v = 2i + 8j - 6k; w = 7i + 4j - 5k

Accepted Answer

A)  48 
B)  -96 
C)  -216 
D)  -80 
A)  48 
B)  -96 
C)  -216 
D)  -80

Question 19

$\text { Express the vector in the form } v = v _ { 1 } i + v _ { 2 } j + v _ { 3 } k \text {. }$
-$$\overrightarrow { \mathrm { AB } } \text { if } \mathrm { A } \text { is the point } ( - 7 , - 6 , - 5 ) \text { and } \mathrm { B } \text { is the point } ( - 2 , - 13 , - 2 )$$

Accepted Answer

A)  v = 5i - 7j - 3k 
B)  v = 5i - 7j + 3k 
C)  v = 5i + 7j - 3k 
D)  v = 5i + 7j + 3k 
A)  v = 5i - 7j - 3k 
B)  v = 5i - 7j + 3k 
C)  v = 5i + 7j - 3k 
D)  v = 5i + 7j + 3k

Question 20

Find the acute angle, in degrees, between the lines.
-$y = \sqrt { 3 } x + 16$ and $y = - \sqrt { 3 } x + 8$

Accepted Answer

A)  $60 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $75 ^ { \circ }$ 
A)  $60 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $75 ^ { \circ }$

Write one or more inequalities that describe the set of points. -The half-space consisting of the points on and behind the yz-plane

Write the equation for the plane. -The plane through the point A(2, 10, 8) perpendicular to the vector from the origin to A.

Find a parametrization for the line segment joining the points. -(2, -7, -2), (0, -7, 4)

Solve the problem. -Find the area of the triangle determined by the points P(-3, 6, -4), Q(-7, , ), and R(-8, -2, ).

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$ - $\mathrm { P } _ { 1 } ( - 4,6,5 )$ and $\mathrm { P } _ { 2 } ( - 1,2,10 )$

Write the equation for the plane. -The plane through the point P(-6, 8, 9) and perpendicular to the line x = 3 + 9t, y = -5 + 9t, z = 6 - t.

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } - \frac { z ^ { 2 } } { 9 } = 1$

Write the equation for the plane. -The plane through the point P(-2, -5, 6) and normal to n = -6i - 3j + 5k.

Solve the problem. -An airplane is flying in the direction $51 ^ { \circ }$ west of north at $660 \mathrm {~km} / \mathrm { hr }$ . Find the component form of the velocity of the airplane, assuming that the positive $x$ -axis represents due east and the positive $y$ -axis represents due north.

Find the center and radius of the sphere. - $x ^ { 2 } + ( y + 10 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 64$

Provide an appropriate response. - $\text { Show that the vectors } | \mathbf { a } | \mathbf { b } + | \mathbf { b } | \mathbf { a } \text { and } | \mathbf { a } | \mathbf { b } - | \mathbf { b } | \mathbf { a } \text { are orthogonal. }$

Write one or more inequalities that describe the set of points. -The exterior of the sphere of radius 5 centered at the point (-2, -3, -5)

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (10, 11); perpendicular to v = 5i - 4j

Find the angle between u and v in radians. -u = 7i + 6j + 3k, v = 5i + 2j + 6k

Find the intersection. -x = 4 + 3t, y = 10 - 10t, z = 2 + 3t ; -2x + 2y - 2z = 9

Find the triple scalar product (u x v) · w of the given vectors. -u = 4i + 2j - k; v = 2i + 8j - 6k; w = 7i + 4j - 5k

$\text { Express the vector in the form } v = v _ { 1 } i + v _ { 2 } j + v _ { 3 } k \text {. }$ - $\overrightarrow { \mathrm { AB } } \text { if } \mathrm { A } \text { is the point } ( - 7 , - 6 , - 5 ) \text { and } \mathrm { B } \text { is the point } ( - 2 , - 13 , - 2 )$

Find the acute angle, in degrees, between the lines. - $y = \sqrt { 3 } x + 16$ and $y = - \sqrt { 3 } x + 8$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 13: Vectors and the Geometry of Space