Question 1

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (9, 4); parallel to v = 2i + 5j

Accepted Answer

A)  $5 x - 2 y = 37$ 
B)  $y - 4 = - \frac { 1 } { 7 } ( x - 2 )$ 
C)  $2 x + 5 y = 38$ 
D)  $2 x + 5 y = 29$ 
A)  $5 x - 2 y = 37$ 
B)  $y - 4 = - \frac { 1 } { 7 } ( x - 2 )$ 
C)  $2 x + 5 y = 38$ 
D)  $2 x + 5 y = 29$

Question 2

Find parametric equations for the line described below.
-The line through the point P(-1, -2, 1) parallel to the vector 5i - 5j - 8k

Accepted Answer

A)  x = -5t- 1, y = -5t - 2, z = 8t + 1 
B)  x = -5t + 1, y = 5t + 2, z = -8t - 1 
C)  x = 5t - 1, y = -5t - 2, z = -8t + 1 
D)  x = 5t + 1, y = -5t + 2, z = -8t - 1 
A)  x = -5t- 1, y = -5t - 2, z = 8t + 1 
B)  x = -5t + 1, y = 5t + 2, z = -8t - 1 
C)  x = 5t - 1, y = -5t - 2, z = -8t + 1 
D)  x = 5t + 1, y = -5t + 2, z = -8t - 1

Question 3

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

Accepted Answer

The answer of Sketch the given surface.
-\[x ^ { 2...

Question 4

Solve the problem.
-Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 1), Q(1, 0, 0) and R(2, 2, 2).

Accepted Answer

A)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { j } + \mathbf { k } )$ 
B)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { j } - \mathbf { k } )$ 
C)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { i } + \mathbf { j } )$ 
D)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { i } - \mathbf { j } )$ 
A)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { j } + \mathbf { k } )$ 
B)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { j } - \mathbf { k } )$ 
C)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { i } + \mathbf { j } )$ 
D)  $\frac { 1 } { \sqrt { 2 } } ( \mathbf { i } - \mathbf { j } )$

Question 5

Find the center and radius of the sphere.
-$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 18 x - 10 y - 6 z = - 15$$

Accepted Answer

A)  C(9, 5, 3), a = 100 
B)  C(-9, -5, -3), a = 10 
C)  C(-9, -5, -3), a = 100 
D)  C(9, 5, 3), a = 10 
A)  C(9, 5, 3), a = 100 
B)  C(-9, -5, -3), a = 10 
C)  C(-9, -5, -3), a = 100 
D)  C(9, 5, 3), a = 10

Question 6

Write one or more inequalities that describe the set of points.
-The rectangular solid in the first octant bounded by the planes x = 5, x = 8, y = 7, y = 9, z = 6 and z = 7 (planes excluded)

Accepted Answer

A)  The given planes do not form a rectangular solid 
B)  x  8; y  9; z  7 
A)  The given planes do not form a rectangular solid 
B)  x  8; y  9; z  7

Question 7

Write the equation for the plane.
-The plane through the points P(5, 2, -30) , Q(-3, 6, -26) and R(-1, -3, 29).

Accepted Answer

A)  4x + 7y + z = -4 
B)  7x + y + 4z = 4 
C)  4x + 7y + z = 4 
D)  7x + y + 4z = -4 
A)  4x + 7y + z = -4 
B)  7x + y + 4z = 4 
C)  4x + 7y + z = 4 
D)  7x + y + 4z = -4

Question 8

Express the vector as a product of its length and direction.
-$\frac { 15 } { 7 } j + \frac { 8 } { 7 } \mathbf { k }$

Accepted Answer

A)  $\frac { 17 } { 7 } \left( \frac { 15 } { 7 } \mathbf { j } + \frac { 8 } { 7 } \mathbf { k } \right)$ 
B)  $\frac { 289 } { 49 } \left( \frac { 15 } { 7 } \mathbf { j } + \frac { 8 } { 7 } \mathbf { k } \right)$ 
C)  $\frac { 17 } { 7 } \left( \frac { 15 } { 17 } \mathbf { j } + \frac { 8 } { 17 } \mathbf { k } \right)$ 
D)  $\frac { 17 } { 7 } ( \mathbf { j } + \mathbf { k } )$ 
A)  $\frac { 17 } { 7 } \left( \frac { 15 } { 7 } \mathbf { j } + \frac { 8 } { 7 } \mathbf { k } \right)$ 
B)  $\frac { 289 } { 49 } \left( \frac { 15 } { 7 } \mathbf { j } + \frac { 8 } { 7 } \mathbf { k } \right)$ 
C)  $\frac { 17 } { 7 } \left( \frac { 15 } { 17 } \mathbf { j } + \frac { 8 } { 17 } \mathbf { k } \right)$ 
D)  $\frac { 17 } { 7 } ( \mathbf { j } + \mathbf { k } )$

Question 9

Find the indicated vector.
-Let $\mathbf { u } = \langle - 2 , - 9 \rangle$. Find $5 \mathbf { u }$.

Accepted Answer

A)  $\langle 10 , - 45 \rangle$ 
B)  $( - 10 , - 45 )$ 
C)  $\langle - 10,45 \rangle$ 
D)  $\langle 10,45 \rangle$ 
A)  $\langle 10 , - 45 \rangle$ 
B)  $( - 10 , - 45 )$ 
C)  $\langle - 10,45 \rangle$ 
D)  $\langle 10,45 \rangle$

Question 10

Match the equation with the surface it defines.
-$$\frac { x ^ { 2 } } { 64 } + \frac { z ^ { 2 } } { 64 } = \frac { y } { 6 }$$

Accepted Answer

A)  Figure 3 
B)  Figure 2 
C)  Figure 1 
D)  Figure 4 
A)  Figure 3 
B)  Figure 2 
C)  Figure 1 
D)  Figure 4

Question 11

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (-5, -5); perpendicular to v = -6i + 6j

Accepted Answer

A)  y + 5 = - 11(x + 6) 
B)  6x + 6y = -60 
C)  -6x + 6y = 72 
D)  -6x + 6y = 0 
A)  y + 5 = - 11(x + 6) 
B)  6x + 6y = -60 
C)  -6x + 6y = 72 
D)  -6x + 6y = 0

Question 12

Find the triple scalar product (u x v) · w of the given vectors.
-u = i + j + k; v = 9i + 7j + 2k; w = 10i + 6j + 5k

Accepted Answer

A)  14 
B)  32 
C)  -18 
D)  106 
A)  14 
B)  32 
C)  -18 
D)  106

Question 13

Calculate the requested distance.
-The distance from the point S(5, -3, -8) to the plane 3x + 4y = -6

Accepted Answer

A)  $\frac { 33 } { 5 }$ 
B)  $\frac { 9 } { 5 }$ 
C)  $\frac { 9 } { 25 }$ 
D)  $\frac { 21 } { 25 }$ 
A)  $\frac { 33 } { 5 }$ 
B)  $\frac { 9 } { 5 }$ 
C)  $\frac { 9 } { 25 }$ 
D)  $\frac { 21 } { 25 }$

Question 14

$\text { Express the vector in the form } v = v _ { 1 } i + v _ { 2 } j + v _ { 3 } k \text {. }$
-$7 \mathbf { u } - 5 \mathbf { v } \text { if } \mathbf { u } = \langle 1,1,0 \rangle \text { and } \mathbf { v } = \langle 3,0,1 \rangle$

Accepted Answer

A)  v = -8i + 12j - 5k 
B)  v = 22i + 7j - 5k 
C)  v = 7i + 7j - 5k 
D)  v = -8i + 7j - 5k 
A)  v = -8i + 12j - 5k 
B)  v = 22i + 7j - 5k 
C)  v = 7i + 7j - 5k 
D)  v = -8i + 7j - 5k

Question 15

Identify the type of surface represented by the given equation.
-$$\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 5 } = \frac { z ^ { 2 } } { 4 }$$

Accepted Answer

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

Question 16

Express the vector as a product of its length and direction.
--5i + 5j + 5k

Accepted Answer

A)  $5 \sqrt { 3 } \left( - \frac { \sqrt { 3 } } { 3 } \mathbf { i } + \frac { \sqrt { 3 } } { 3 } \mathbf { j } + \frac { \sqrt { 3 } } { 3 } \mathbf { k } \right)$ 
B)  $\frac { \sqrt { 3 } } { 15 } \left( - \frac { \sqrt { 3 } } { 3 } \mathbf { i } + \frac { \sqrt { 3 } } { 3 } \mathbf { j } + \frac { \sqrt { 3 } } { 3 } \mathbf { k } \right)$ 
C)  $5 \sqrt { 3 } \left( - \frac { 1 } { 75 } \mathbf { i } + \frac { 1 } { 75 } \mathbf { j } + \frac { 1 } { 75 } \mathbf { k } \right)$ 
D)  $5 ( - \mathbf { i } + \mathbf { j } + \mathbf { k } )$ 
A)  $5 \sqrt { 3 } \left( - \frac { \sqrt { 3 } } { 3 } \mathbf { i } + \frac { \sqrt { 3 } } { 3 } \mathbf { j } + \frac { \sqrt { 3 } } { 3 } \mathbf { k } \right)$ 
B)  $\frac { \sqrt { 3 } } { 15 } \left( - \frac { \sqrt { 3 } } { 3 } \mathbf { i } + \frac { \sqrt { 3 } } { 3 } \mathbf { j } + \frac { \sqrt { 3 } } { 3 } \mathbf { k } \right)$ 
C)  $5 \sqrt { 3 } \left( - \frac { 1 } { 75 } \mathbf { i } + \frac { 1 } { 75 } \mathbf { j } + \frac { 1 } { 75 } \mathbf { k } \right)$ 
D)  $5 ( - \mathbf { i } + \mathbf { j } + \mathbf { k } )$

Question 17

Find the intersection.
-x + y + z = , x + y =

Accepted Answer

A)  x = -t, y = 13 + t, z = -8 
B)  x = -t, y = 13 + t, z = 8 
C)  x = t, y = 13 - t, z = -8 
D)  x = -1, y = 1 + 13t, z = -8t 
A)  x = -t, y = 13 + t, z = -8 
B)  x = -t, y = 13 + t, z = 8 
C)  x = t, y = 13 - t, z = -8 
D)  x = -1, y = 1 + 13t, z = -8t

Question 18

Solve the problem.
-A ramp leading to the entrance of a building is inclined upward at an angle of $3 ^ { \circ }$. A suitcase is to be pulled up the ramp by a handle that makes an angle of $38 ^ { \circ }$ with the horizontal. How much force must be applied in the direction of the handle so that the component of the force parallel to the ramp is 50 lbs.?

Accepted Answer

A)  40.96 lbs 
B)  59.05 lbs 
C)  63.36 lbs 
D)  -51.83 lbs 
A)  40.96 lbs 
B)  59.05 lbs 
C)  63.36 lbs 
D)  -51.83 lbs

Question 19

Find the acute angle, in degrees, between the lines.
-$( 1 + \sqrt { 3 } ) x + ( 1 - \sqrt { 3 } ) y = 10$ and $\sqrt { 3 } x + y = 14$

Accepted Answer

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

Question 20

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$
-$\mathrm { P } _ { 1 } ( 7,8,5 )$ and $\mathrm { P } _ { 2 } ( 15,16,9 )$

Accepted Answer

A)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } + \frac { 1 } { 3 } \mathbf { k } ; ( 11,12,7 )$ 
B)  $\frac { 7 } { 3 } \mathbf { i } + \frac { 8 } { 3 } \mathbf { j } + \frac { 5 } { 3 } \mathbf { k } ; \left( \frac { 7 } { 2 } , 4 , \frac { 5 } { 2 } \right)$ 
C)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 1 } { 3 } \mathbf { k } ; ( 4,4,2 )$ 
D)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 1 } { 3 } \mathbf { k } ; \left( \frac { 15 } { 2 } , 8 , \frac { 9 } { 2 } \right)$ 
A)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } + \frac { 1 } { 3 } \mathbf { k } ; ( 11,12,7 )$ 
B)  $\frac { 7 } { 3 } \mathbf { i } + \frac { 8 } { 3 } \mathbf { j } + \frac { 5 } { 3 } \mathbf { k } ; \left( \frac { 7 } { 2 } , 4 , \frac { 5 } { 2 } \right)$ 
C)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 1 } { 3 } \mathbf { k } ; ( 4,4,2 )$ 
D)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 1 } { 3 } \mathbf { k } ; \left( \frac { 15 } { 2 } , 8 , \frac { 9 } { 2 } \right)$

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (9, 4); parallel to v = 2i + 5j

Find parametric equations for the line described below. -The line through the point P(-1, -2, 1) parallel to the vector 5i - 5j - 8k

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

Solve the problem. -Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 1), Q(1, 0, 0) and R(2, 2, 2).

Find the center and radius of the sphere. - $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 18 x - 10 y - 6 z = - 15$

Write one or more inequalities that describe the set of points. -The rectangular solid in the first octant bounded by the planes x = 5, x = 8, y = 7, y = 9, z = 6 and z = 7 (planes excluded)

Write the equation for the plane. -The plane through the points P(5, 2, -30) , Q(-3, 6, -26) and R(-1, -3, 29).

Express the vector as a product of its length and direction. - $\frac { 15 } { 7 } j + \frac { 8 } { 7 } \mathbf { k }$

Find the indicated vector. -Let $\mathbf { u } = \langle - 2 , - 9 \rangle$ . Find $5 \mathbf { u }$ .

Match the equation with the surface it defines. - $\frac { x ^ { 2 } } { 64 } + \frac { z ^ { 2 } } { 64 } = \frac { y } { 6 }$ $Match the equation with the surface it defines. - \frac { x ^ { 2 } } { 64 } + \frac { z ^ { 2 } } { 64 } = \frac { y } { 6 }$

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (-5, -5); perpendicular to v = -6i + 6j

Find the triple scalar product (u x v) · w of the given vectors. -u = i + j + k; v = 9i + 7j + 2k; w = 10i + 6j + 5k

Calculate the requested distance. -The distance from the point S(5, -3, -8) to the plane 3x + 4y = -6

$\text { Express the vector in the form } v = v _ { 1 } i + v _ { 2 } j + v _ { 3 } k \text {. }$ - $7 \mathbf { u } - 5 \mathbf { v } \text { if } \mathbf { u } = \langle 1,1,0 \rangle \text { and } \mathbf { v } = \langle 3,0,1 \rangle$

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 5 } = \frac { z ^ { 2 } } { 4 }$

Express the vector as a product of its length and direction. --5i + 5j + 5k

Find the intersection. -x + y + z = , x + y =

Find the acute angle, in degrees, between the lines. - $( 1 + \sqrt { 3 } ) x + ( 1 - \sqrt { 3 } ) y = 10$ and $\sqrt { 3 } x + y = 14$

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$ - $\mathrm { P } _ { 1 } ( 7,8,5 )$ and $\mathrm { P } _ { 2 } ( 15,16,9 )$

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Exam 13: Vectors and the Geometry of Space