Question 1

Write one or more inequalities that describe the set of points.
-The interior of the sphere $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } = 36$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } \leq 36$ 
B)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } \leq 6$ 
C)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } < 36$ 
D)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } < 6$ 
A)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } \leq 36$ 
B)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } \leq 6$ 
C)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } < 36$ 
D)  $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } < 6$

Question 2

Solve the problem.
-How much work does it take to slide a box 12 meters along the ground by pulling it with a $180 \mathrm {~N}$ force at an angle of $45 ^ { \circ }$ from the horizontal?

Accepted Answer

A)  2160 joules 
B)  $\frac { 2160 } { \sqrt { 2 } }$ joules 
C)  $1080 \sqrt { 2 }$ joules 
D)  $2160 \sqrt { 2 }$ joules 
A)  2160 joules 
B)  $\frac { 2160 } { \sqrt { 2 } }$ joules 
C)  $1080 \sqrt { 2 }$ joules 
D)  $2160 \sqrt { 2 }$ joules

Question 3

Find the length and direction (when defined) of u × v.
-u = 2i + 2j - k, v = -i + k

Accepted Answer

A)  $3 ; \frac { 2 } { 3 } \mathrm { i } - \frac { 1 } { 3 } \mathrm { j } + \frac { 2 } { 3 } \mathrm { k }$ 
B)  $9 ; \frac { 2 } { 9 } \mathrm { i } + \frac { 1 } { 9 } \mathrm { j } - \frac { 2 } { 9 } \mathrm { k }$ 
C)  $3 ; \frac { 2 } { 3 } \mathrm { i } + \frac { 1 } { 3 } \mathrm { j } - \frac { 2 } { 3 } \mathrm { k }$ 
D)  $9 ; \frac { 2 } { 9 } \mathrm { i } - \frac { 1 } { 9 } \mathrm { j } + \frac { 2 } { 9 } \mathrm { k }$ 
A)  $3 ; \frac { 2 } { 3 } \mathrm { i } - \frac { 1 } { 3 } \mathrm { j } + \frac { 2 } { 3 } \mathrm { k }$ 
B)  $9 ; \frac { 2 } { 9 } \mathrm { i } + \frac { 1 } { 9 } \mathrm { j } - \frac { 2 } { 9 } \mathrm { k }$ 
C)  $3 ; \frac { 2 } { 3 } \mathrm { i } + \frac { 1 } { 3 } \mathrm { j } - \frac { 2 } { 3 } \mathrm { k }$ 
D)  $9 ; \frac { 2 } { 9 } \mathrm { i } - \frac { 1 } { 9 } \mathrm { j } + \frac { 2 } { 9 } \mathrm { k }$

Question 4

Write one or more inequalities that describe the set of points.
-The slab bounded by the planes x = and x = 5 (planes included)

Accepted Answer

A)  $4 \leq x \leq 5$ 
B)  $x = 4$ and $x = 5$ 
C)  $- \infty < y < \infty$ and $- \infty < z < \infty$ 
D)  $x \leq 4$ and $x \geq 5$ 
A)  $4 \leq x \leq 5$ 
B)  $x = 4$ and $x = 5$ 
C)  $- \infty < y < \infty$ and $- \infty < z < \infty$ 
D)  $x \leq 4$ and $x \geq 5$

Question 5

Find v · u.
-$\mathbf { v } = \left\langle \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 7 } } \right\rangle$ and $\mathbf { u } = \left\langle \frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 7 } } \right|$

Accepted Answer

A)  $\frac { 1 } { 2 } \mathrm { i } - \frac { 1 } { 7 } \mathrm { j }$ 
B)  0 
C)  $\frac { 2 } { \sqrt { 2 } } \mathrm { i }$ 
D)  $\frac { 5 } { 14 }$ 
A)  $\frac { 1 } { 2 } \mathrm { i } - \frac { 1 } { 7 } \mathrm { j }$ 
B)  0 
C)  $\frac { 2 } { \sqrt { 2 } } \mathrm { i }$ 
D)  $\frac { 5 } { 14 }$

Question 6

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (9, 2); perpendicular to v = 4i + 4j

Accepted Answer

A)  $4 x + 4 y = 44$ 
B)  $4 x - 4 y = 28$ 
C)  $4 x + 4 y = 32$ 
D)  $y - 2 = - \frac { 2 } { 5 } ( x - 4 )$ 
A)  $4 x + 4 y = 44$ 
B)  $4 x - 4 y = 28$ 
C)  $4 x + 4 y = 32$ 
D)  $y - 2 = - \frac { 2 } { 5 } ( x - 4 )$

Question 7

Find v · u.
-$\mathbf { v } = 5 \mathbf { i } - 9 \mathbf { j }$ and $\mathbf { u } = - 3 \sqrt { 3 } \mathbf { i } + 7 \mathbf { j }$

Accepted Answer

A)  $( 5 - 3 \sqrt { 3 } ) \mathbf { i } - 2 \mathrm { j }$ 
B)  $- 63 \sqrt { 3 } \mathrm { i } - 15 \mathrm { j }$ 
C)  $- 63 \sqrt { 3 } + 15$ 
D)  $- 63 - 15 \sqrt { 3 }$ 
A)  $( 5 - 3 \sqrt { 3 } ) \mathbf { i } - 2 \mathrm { j }$ 
B)  $- 63 \sqrt { 3 } \mathrm { i } - 15 \mathrm { j }$ 
C)  $- 63 \sqrt { 3 } + 15$ 
D)  $- 63 - 15 \sqrt { 3 }$

Question 8

Sketch the given surface.
-$$16 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$$

Accepted Answer

The answer of Sketch the given surface.
-\[16 x ^ {...

Question 9

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
-8x - 10y + 2z = -10 and -4x - 6y + 5z = -7

Accepted Answer

A)  0.341 rad 
B)  1.230 rad 
C)  1.571 rad 
D)  0.459 rad 
A)  0.341 rad 
B)  1.230 rad 
C)  1.571 rad 
D)  0.459 rad

Question 10

Find the acute angle, in radians, between the lines.
-3x - 4y = 2 and 5x + 4y = -5

Accepted Answer

A)  3.110 radians 
B)  0.03124 radians 
C)  1.540 radians 
D)  0.03123 radians 
A)  3.110 radians 
B)  0.03124 radians 
C)  1.540 radians 
D)  0.03123 radians

Question 11

Find v · u.
-v = -4i + 9j and u = 6i + 5j

Accepted Answer

A)  -24i + 45j 
B)  21 
C)  -69 
D)  2i + 14j 
A)  -24i + 45j 
B)  21 
C)  -69 
D)  2i + 14j

Question 12

Find an equation for the sphere with the given center and radius.
-Center (0, -8, -1), radius = 10

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } + 10 z ^ { 2 } + 16 y + 2 z = 35$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 y + 2 z = 35$ 
C)  $x ^ { 2 } + y ^ { 2 } + 10 z ^ { 2 } + 16 y - 2 z = 35$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 y - 2 z = 35$ 
A)  $x ^ { 2 } + y ^ { 2 } + 10 z ^ { 2 } + 16 y + 2 z = 35$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 y + 2 z = 35$ 
C)  $x ^ { 2 } + y ^ { 2 } + 10 z ^ { 2 } + 16 y - 2 z = 35$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 y - 2 z = 35$

Question 13

Find parametric equations for the line described below.
-The line through the points P(-1, -1, -2) and Q(-5, -4, )

Accepted Answer

A)  x = t - 4, y = t - 3, z = -2t + 3 
B)  x = -4t + 1, y = -3t + 1, z = 3t + 2 
C)  x = -4t - 1, y = -3t - 1, z = 3t - 2 
D)  x = t + 4, y = t + 3, z = -2t - 3 
A)  x = t - 4, y = t - 3, z = -2t + 3 
B)  x = -4t + 1, y = -3t + 1, z = 3t + 2 
C)  x = -4t - 1, y = -3t - 1, z = 3t - 2 
D)  x = t + 4, y = t + 3, z = -2t - 3

Question 14

Identify the type of surface represented by the given equation.
-$$\frac { z ^ { 2 } } { 3 } - \frac { x ^ { 2 } } { 7 } = \frac { y } { 10 }$$

Accepted Answer

A)  Parabolic cylinder 
B)  Ellipsoid 
C)  Paraboloid 
D)  Hyperbolic paraboloid 
A)  Parabolic cylinder 
B)  Ellipsoid 
C)  Paraboloid 
D)  Hyperbolic paraboloid

Question 15

Describe the given set of points with a single equation or with a pair of equations.
-The circle in which the plane through the point (-8, 12, ) perpendicular to the y-axis meets the sphere of radius 15 centered at the origin.

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 225$ and $y = 12$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81$ and $y = 12$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 63$ and $y = 12$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 144$ and $y = 12$ 
A)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 225$ and $y = 12$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81$ and $y = 12$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 63$ and $y = 12$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 144$ and $y = 12$

Question 16

Find parametric equations for the line described below.
-The line through the point P(-2, , -5) and perpendicular to the vectors u = i - 5j + 7k and v = -6i - 3j + 4k

Accepted Answer

A)  x = 1t - 2, y = -62t + 5, z = -45t - 5 
B)  x = 1t - 2, y = -62t + 5, z = 5t - 5 
C)  x = 1t - 2, y = 62t + 5, z = -45t - 5 
D)  x = 1t + 2, y = -62t - 5, z = 5t + 5 
A)  x = 1t - 2, y = -62t + 5, z = -45t - 5 
B)  x = 1t - 2, y = -62t + 5, z = 5t - 5 
C)  x = 1t - 2, y = 62t + 5, z = -45t - 5 
D)  x = 1t + 2, y = -62t - 5, z = 5t + 5

Question 17

Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin.
-u = i + k, v = i - k

Accepted Answer

The answer of Sketch the coordinate axes and then include...

Question 18

Find the acute angle, in degrees, between the lines.
-$x - \sqrt { 3 } y = 6$ and $\sqrt { 3 } x - y = 1$

Accepted Answer

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

Question 19

Write the equation for the plane.
-The plane through the point A(-5, -2, 4) perpendicular to the vector from the origin to A.

Accepted Answer

A)  $- 5 x - 2 y + 4 z = - 45$ 
B)  $5 x + 2 y - 4 z = - 45$ 
C)  $5 x + 2 y - 4 z = \sqrt { 45 }$ 
D)  $5 x - 2 y - 4 z = 3$ 
A)  $- 5 x - 2 y + 4 z = - 45$ 
B)  $5 x + 2 y - 4 z = - 45$ 
C)  $5 x + 2 y - 4 z = \sqrt { 45 }$ 
D)  $5 x - 2 y - 4 z = 3$

Question 20

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$
-$\mathrm { P } _ { 1 } ( - 2,8 , - 1 )$ and $\mathrm { P } _ { 2 } ( 0,2,2 )$

Accepted Answer

A)  $\frac { 2 } { \sqrt { 7 } } \mathbf { i } + \frac { 8 } { \sqrt { 7 } } \mathbf { j } + \frac { 1 } { \sqrt { 7 } } \mathbf { k } ; \left( - 1,4 , - \frac { 1 } { 2 } \right)$ 
B)  $\frac { 2 } { 7 } \mathbf { i } - \frac { 6 } { 7 } \mathbf { j } + \frac { 3 } { 7 } \mathbf { k } ; ( 0,1,1 )$ 
C)  $\frac { 2 } { 7 } \mathbf { i } - \frac { 6 } { 7 } \mathrm { j } + \frac { 3 } { 7 } \mathbf { k } ; \left( - 1,5 , \frac { 1 } { 2 } \right)$ 
D)  $\frac { 2 } { \sqrt { 7 } } \mathbf { i } - \frac { 6 } { \sqrt { 7 } } \mathbf { j } + \frac { 3 } { \sqrt { 7 } } \mathbf { k } ; \left( - 1,5 , \frac { 1 } { 2 } \right)$ 
A)  $\frac { 2 } { \sqrt { 7 } } \mathbf { i } + \frac { 8 } { \sqrt { 7 } } \mathbf { j } + \frac { 1 } { \sqrt { 7 } } \mathbf { k } ; \left( - 1,4 , - \frac { 1 } { 2 } \right)$ 
B)  $\frac { 2 } { 7 } \mathbf { i } - \frac { 6 } { 7 } \mathbf { j } + \frac { 3 } { 7 } \mathbf { k } ; ( 0,1,1 )$ 
C)  $\frac { 2 } { 7 } \mathbf { i } - \frac { 6 } { 7 } \mathrm { j } + \frac { 3 } { 7 } \mathbf { k } ; \left( - 1,5 , \frac { 1 } { 2 } \right)$ 
D)  $\frac { 2 } { \sqrt { 7 } } \mathbf { i } - \frac { 6 } { \sqrt { 7 } } \mathbf { j } + \frac { 3 } { \sqrt { 7 } } \mathbf { k } ; \left( - 1,5 , \frac { 1 } { 2 } \right)$

Write one or more inequalities that describe the set of points. -The interior of the sphere $x ^ { 2 } + y ^ { 2 } + x ^ { 2 } = 36$

Solve the problem. -How much work does it take to slide a box 12 meters along the ground by pulling it with a $180 \mathrm {~N}$ force at an angle of $45 ^ { \circ }$ from the horizontal?

Find the length and direction (when defined) of u × v. -u = 2i + 2j - k, v = -i + k

Write one or more inequalities that describe the set of points. -The slab bounded by the planes x = and x = 5 (planes included)

Find v · u. - $\mathbf { v } = \left\langle \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 7 } } \right\rangle$ and $\mathbf { u } = \left\langle \frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 7 } } \right|$

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (9, 2); perpendicular to v = 4i + 4j

Find v · u. - $\mathbf { v } = 5 \mathbf { i } - 9 \mathbf { j }$ and $\mathbf { u } = - 3 \sqrt { 3 } \mathbf { i } + 7 \mathbf { j }$

Sketch the given surface. - $16 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ $Sketch the given surface. - 16 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. -8x - 10y + 2z = -10 and -4x - 6y + 5z = -7

Find the acute angle, in radians, between the lines. -3x - 4y = 2 and 5x + 4y = -5

Find v · u. -v = -4i + 9j and u = 6i + 5j

Find an equation for the sphere with the given center and radius. -Center (0, -8, -1), radius = 10

Find parametric equations for the line described below. -The line through the points P(-1, -1, -2) and Q(-5, -4, )

Identify the type of surface represented by the given equation. - $\frac { z ^ { 2 } } { 3 } - \frac { x ^ { 2 } } { 7 } = \frac { y } { 10 }$

Describe the given set of points with a single equation or with a pair of equations. -The circle in which the plane through the point (-8, 12, ) perpendicular to the y-axis meets the sphere of radius 15 centered at the origin.

Find parametric equations for the line described below. -The line through the point P(-2, , -5) and perpendicular to the vectors u = i - 5j + 7k and v = -6i - 3j + 4k

Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin. -u = i + k, v = i - k

Find the acute angle, in degrees, between the lines. - $x - \sqrt { 3 } y = 6$ and $\sqrt { 3 } x - y = 1$

Write the equation for the plane. -The plane through the point A(-5, -2, 4) perpendicular to the vector from the origin to A.

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$ - $\mathrm { P } _ { 1 } ( - 2,8 , - 1 )$ and $\mathrm { P } _ { 2 } ( 0,2,2 )$

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Exam 13: Vectors and the Geometry of Space

Write one or more inequalities that describe the set of points. -The interior of the sphere x2+y2+x2=36x ^ { 2 } + y ^ { 2 } + x ^ { 2 } = 36x2+y2+x2=36

Solve the problem. -How much work does it take to slide a box 12 meters along the ground by pulling it with a 180 N180 \mathrm {~N}180 N force at an angle of 45∘45 ^ { \circ }45∘ from the horizontal?