Question 1

Identify the type of surface represented by the given equation.
-$$\frac { x ^ { 2 } } { 9 } + \frac { z ^ { 2 } } { 9 } = \frac { y } { 5 }$$

Accepted Answer

A)  Elliptical cone 
B)  Ellipsoid 
C)  Hyperbolic paraboloid 
D)  Elliptical paraboloid 
A)  Elliptical cone 
B)  Ellipsoid 
C)  Hyperbolic paraboloid 
D)  Elliptical paraboloid

Question 2

Determine whether the following is always true or not always true. Given reasons for your answers.
-u × (v + w) = u × v + u × w

Accepted Answer

Always tru

Question 3

Find v · u.
-v = 3i + 8j and u = 7i + 3j

Accepted Answer

A)  10i + 11j 
B)  45 
C)  -3 
D)  21i + 24j 
A)  10i + 11j 
B)  45 
C)  -3 
D)  21i + 24j

Question 4

Find the acute angle, in radians, between the lines.
-7x - 5y = 4 and 4x - 5y = -8

Accepted Answer

A)  0.9622 radians 
B)  1.847 radians 
C)  1.295 radians 
D)  0.2758 radians 
A)  0.9622 radians 
B)  1.847 radians 
C)  1.295 radians 
D)  0.2758 radians

Question 5

Find parametric equations for the line described below.
-The line through the point P(-2, 7, -7) and perpendicular to the plane 1x + 4y + 2z = 5

Accepted Answer

A)  x = 1t - 2, y = 4t + 7, z = 2t - 7 
B)  x = -1t + 2, y = -4t - 7, z = -2t + 7 
C)  x = 1t + 2, y = 4t - 7, z = 2t + 7 
D)  x = -4t - 2, y = 1t + 7, z = -7 
A)  x = 1t - 2, y = 4t + 7, z = 2t - 7 
B)  x = -1t + 2, y = -4t - 7, z = -2t + 7 
C)  x = 1t + 2, y = 4t - 7, z = 2t + 7 
D)  x = -4t - 2, y = 1t + 7, z = -7

Question 6

Solve the problem.
-Show that the point P( , -5, ) is equidistant from the points A( , -7, ) and B( , -3, ).

Accepted Answer

A)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is $\sqrt { 6 }$; the distance between $\mathrm { P }$ and $\mathrm { B }$ is $\sqrt { 6 }$ 
B)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is $\sqrt { 5 }$; the distance between $\mathrm { P }$ and $\mathrm { B }$ is $\sqrt { 5 }$ 
C)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is 2 ; the distance between $\mathrm { P }$ and $\mathrm { B }$ is 2 
D)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is 3 ; the distance between $\mathrm { P }$ and $\mathrm { B }$ is 3 
A)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is $\sqrt { 6 }$; the distance between $\mathrm { P }$ and $\mathrm { B }$ is $\sqrt { 6 }$ 
B)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is $\sqrt { 5 }$; the distance between $\mathrm { P }$ and $\mathrm { B }$ is $\sqrt { 5 }$ 
C)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is 2 ; the distance between $\mathrm { P }$ and $\mathrm { B }$ is 2 
D)  The distance between $\mathrm { P }$ and $\mathrm { A }$ is 3 ; the distance between $\mathrm { P }$ and $\mathrm { B }$ is 3

Question 7

$$\text { Use the vectors } u , v , w \text {, and } z \text { head to tail as needed to sketch the indicated vector. }$$

-$\mathbf { u } + \mathbf { z }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 8

Solve the problem.
-Find a vector of magnitude 11 in the direction of v = 5i - 12k.

Accepted Answer

A)  $\frac { 11 } { 13 } ( 5 \mathbf { i } - 12 \mathbf { k } )$ 
B)  $11 ( 5 \mathbf { i } - 12 \mathbf { k } )$ 
C)  $11 ( \mathbf { i } - \mathbf { k } )$ 
D)  $\frac { 1 } { 13 } ( 5 \mathbf { i } - 12 \mathbf { k } )$ 
A)  $\frac { 11 } { 13 } ( 5 \mathbf { i } - 12 \mathbf { k } )$ 
B)  $11 ( 5 \mathbf { i } - 12 \mathbf { k } )$ 
C)  $11 ( \mathbf { i } - \mathbf { k } )$ 
D)  $\frac { 1 } { 13 } ( 5 \mathbf { i } - 12 \mathbf { k } )$

Question 9

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

Accepted Answer

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

Question 10

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (8, 12); parallel to v = 6i - 6j

Accepted Answer

A)  6x - 6y = -24 
B)  -6x - 6y = -120 
C)  6x - 6y = 72 
D)  y - 12 = 9(x - 6) 
A)  6x - 6y = -24 
B)  -6x - 6y = -120 
C)  6x - 6y = 72 
D)  y - 12 = 9(x - 6)

Question 11

Determine whether the following is always true or not always true. Given reasons for your answers.
-$|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}}$

Accepted Answer

Always tru

Question 12

Find the center and radius of the sphere.
-$$2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - x + y - z = 9$$

Accepted Answer

A)  $\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 75 } { 8 }$ 
B)  $\left( \frac { 1 } { 4 } , - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 5 \sqrt { 6 } } { 4 }$ 
C)  $\left( \frac { 1 } { 4 } , - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 5 \sqrt { 6 } } { 2 }$ 
D)  $\left( \frac { 1 } { 4 } , - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 5 \sqrt { 3 } } { 4 }$ 
A)  $\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 75 } { 8 }$ 
B)  $\left( \frac { 1 } { 4 } , - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 5 \sqrt { 6 } } { 4 }$ 
C)  $\left( \frac { 1 } { 4 } , - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 5 \sqrt { 6 } } { 2 }$ 
D)  $\left( \frac { 1 } { 4 } , - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) , \mathrm { a } = \frac { 5 \sqrt { 3 } } { 4 }$

Question 13

Find the length and direction (when defined) of u × v.
-u = 8i + 5j , v = i - j

Accepted Answer

A)  3; 3k 
B)  3; -3k 
C)  13; -13k 
D)  13; -k 
A)  3; 3k 
B)  3; -3k 
C)  13; -13k 
D)  13; -k

Question 14

Find the center and radius of the sphere.
-$$3 x ^ { 2 } + 3 y ^ { 2 } + 3 z ^ { 2 } - 2 x + 2 y = 9$$

Accepted Answer

A)  $\left( \frac { 1 } { 3 } , - \frac { 1 } { 3 } , 0 \right) , \mathrm { a } = \frac { \sqrt { 29 } } { 3 }$ 
B)  $\left( - \frac { 1 } { 9 } , \frac { 1 } { 9 } , 0 \right) , a = \frac { \sqrt { 29 } } { 9 }$ 
C)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } , 0 \right) , a = \frac { 29 } { 9 }$ 
D)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } , 0 \right) , \mathrm { a } = \frac { \sqrt { 29 } } { 3 }$ 
A)  $\left( \frac { 1 } { 3 } , - \frac { 1 } { 3 } , 0 \right) , \mathrm { a } = \frac { \sqrt { 29 } } { 3 }$ 
B)  $\left( - \frac { 1 } { 9 } , \frac { 1 } { 9 } , 0 \right) , a = \frac { \sqrt { 29 } } { 9 }$ 
C)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } , 0 \right) , a = \frac { 29 } { 9 }$ 
D)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } , 0 \right) , \mathrm { a } = \frac { \sqrt { 29 } } { 3 }$

Question 15

Sketch the given surface.
-$$z = x ^ { 2 } + 4 y ^ { 2 }$$

Accepted Answer

The answer of Sketch the given surface.
-\[z = x ^...

Question 16

Find the length and direction (when defined) of u × v.
-u = 4i + 2j + 8k, v = -i - 2j - 2k

Accepted Answer

A)  $6 \sqrt { 5 } ; \frac { 2 \sqrt { 5 } } { 5 } \mathbf { i } + \frac { \sqrt { 5 } } { 5 } \mathbf { k }$ 
B)  $180 ; \frac { 2 \sqrt { 5 } } { 15 } \mathbf { i } + \frac { \sqrt { 15 } } { 15 } \mathbf { j } + \frac { \sqrt { 5 } } { 15 } \mathbf { k }$ 
C)  $6 \sqrt { 5 } ; \frac { 2 \sqrt { 5 } } { 5 } \mathbf { i } - \frac { \sqrt { 5 } } { 5 } \mathbf { k }$ 
D)  $180 ; \frac { 1 } { 15 } \mathbf { i } + \frac { 1 } { 30 } \mathbf { k }$ 
A)  $6 \sqrt { 5 } ; \frac { 2 \sqrt { 5 } } { 5 } \mathbf { i } + \frac { \sqrt { 5 } } { 5 } \mathbf { k }$ 
B)  $180 ; \frac { 2 \sqrt { 5 } } { 15 } \mathbf { i } + \frac { \sqrt { 15 } } { 15 } \mathbf { j } + \frac { \sqrt { 5 } } { 15 } \mathbf { k }$ 
C)  $6 \sqrt { 5 } ; \frac { 2 \sqrt { 5 } } { 5 } \mathbf { i } - \frac { \sqrt { 5 } } { 5 } \mathbf { k }$ 
D)  $180 ; \frac { 1 } { 15 } \mathbf { i } + \frac { 1 } { 30 } \mathbf { k }$

Question 17

Solve the problem.
-Find the vector from the origin to the center of mass of a thin triangular plate (uniform density) whose vertices are A(7, 9, 10), B(8, 3, 8), and C(9, 2, 9).

Accepted Answer

A)  $- 1 \mathbf { i } + \frac { 8 } { 3 } j + \frac { 1 } { 3 } \mathbf { k }$ 
B)  $- \frac { 1 } { 2 } \mathbf { i } + \frac { 4 } { 3 } \mathrm { j } + \frac { 1 } { 6 } \mathbf { k }$ 
C)  $8 i + \frac { 14 } { 3 } j + 9 k$ 
D)  $- \frac { 3 } { 2 } \mathbf { i } + 4 \mathbf { j } + \frac { 1 } { 2 } \mathbf { k }$ 
A)  $- 1 \mathbf { i } + \frac { 8 } { 3 } j + \frac { 1 } { 3 } \mathbf { k }$ 
B)  $- \frac { 1 } { 2 } \mathbf { i } + \frac { 4 } { 3 } \mathrm { j } + \frac { 1 } { 6 } \mathbf { k }$ 
C)  $8 i + \frac { 14 } { 3 } j + 9 k$ 
D)  $- \frac { 3 } { 2 } \mathbf { i } + 4 \mathbf { j } + \frac { 1 } { 2 } \mathbf { k }$

Question 18

Find the length and direction (when defined) of u × v.
-u = 3i + 5j - 3k, v = -6i - 10j + 6k

Accepted Answer

A)  $\sqrt { 215 } ; \frac { 1 } { \sqrt { 215 } } ( - 3 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } )$ 
B)  $\sqrt { 129 } ; \frac { 1 } { \sqrt { 129 } } ( - 3 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } )$ 
C)  $0 ; - 3 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k }$ 
D)  0 ; no direction 
A)  $\sqrt { 215 } ; \frac { 1 } { \sqrt { 215 } } ( - 3 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } )$ 
B)  $\sqrt { 129 } ; \frac { 1 } { \sqrt { 129 } } ( - 3 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } )$ 
C)  $0 ; - 3 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k }$ 
D)  0 ; no direction

Question 19

Give a geometric description of the set of points whose coordinates satisfy the given conditions.
-$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } > 9$$

Accepted Answer

A)  All points outside the cylinder with radius 3 
B)  All points outside the sphere of radius 3 
C)  All points in space 
D)  All points on the surface of the cylinder with radius 3 
A)  All points outside the cylinder with radius 3 
B)  All points outside the sphere of radius 3 
C)  All points in space 
D)  All points on the surface of the cylinder with radius 3

Question 20

Find a parametrization for the line segment joining the points.
-(4, 0, 3), (0, 4, 0)

Accepted Answer

A)  $x = - 3 t , y = - 3 t + 4 , z = - 2 t , 0 \leq t \leq 1$ 
B)  $x = - 3 t + 4 , y = - 3 t , z = - 2 t + 3,0 \leq t \leq 1$ 
C)  $x = 4 t , y = - 4 t + 4 , z = 3 t , 0 \leq t \leq 1$ 
D)  $x = - 4 t + 4 , y = 4 t , z = - 3 t + 3,0 \leq t \leq 1$ 
A)  $x = - 3 t , y = - 3 t + 4 , z = - 2 t , 0 \leq t \leq 1$ 
B)  $x = - 3 t + 4 , y = - 3 t , z = - 2 t + 3,0 \leq t \leq 1$ 
C)  $x = 4 t , y = - 4 t + 4 , z = 3 t , 0 \leq t \leq 1$ 
D)  $x = - 4 t + 4 , y = 4 t , z = - 3 t + 3,0 \leq t \leq 1$

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 9 } + \frac { z ^ { 2 } } { 9 } = \frac { y } { 5 }$

Determine whether the following is always true or not always true. Given reasons for your answers. -u × (v + w) = u × v + u × w

Find v · u. -v = 3i + 8j and u = 7i + 3j

Find the acute angle, in radians, between the lines. -7x - 5y = 4 and 4x - 5y = -8

Find parametric equations for the line described below. -The line through the point P(-2, 7, -7) and perpendicular to the plane 1x + 4y + 2z = 5

Solve the problem. -Show that the point P( , -5, ) is equidistant from the points A( , -7, ) and B( , -3, ).

Solve the problem. -Find a vector of magnitude 11 in the direction of v = 5i - 12k.

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

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (8, 12); parallel to v = 6i - 6j

Determine whether the following is always true or not always true. Given reasons for your answers. - $|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}}$

Find the center and radius of the sphere. - $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - x + y - z = 9$

Find the length and direction (when defined) of u × v. -u = 8i + 5j , v = i - j

Find the center and radius of the sphere. - $3 x ^ { 2 } + 3 y ^ { 2 } + 3 z ^ { 2 } - 2 x + 2 y = 9$

Sketch the given surface. - $z = x ^ { 2 } + 4 y ^ { 2 }$ $Sketch the given surface. - z = x ^ { 2 } + 4 y ^ { 2 }$

Find the length and direction (when defined) of u × v. -u = 4i + 2j + 8k, v = -i - 2j - 2k

Solve the problem. -Find the vector from the origin to the center of mass of a thin triangular plate (uniform density) whose vertices are A(7, 9, 10), B(8, 3, 8), and C(9, 2, 9).

Find the length and direction (when defined) of u × v. -u = 3i + 5j - 3k, v = -6i - 10j + 6k

Give a geometric description of the set of points whose coordinates satisfy the given conditions. - $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } > 9$

Find a parametrization for the line segment joining the points. -(4, 0, 3), (0, 4, 0)

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Exam 13: Vectors and the Geometry of Space

Identify the type of surface represented by the given equation. - x29+z29=y5\frac { x ^ { 2 } } { 9 } + \frac { z ^ { 2 } } { 9 } = \frac { y } { 5 }9x2​+9z2​=5y​