Question 1

Calculate the requested distance.
-The distance from the point S(1, -7, -9) to the plane 10x + 11y + 2z = -8

Accepted Answer

A)  $\frac { 77 } { 15 }$ 
B)  $\frac { 77 } { 225 }$ 
C)  $\frac { 19 } { 75 }$ 
D)  $\frac { 19 } { 5 }$ 
A)  $\frac { 77 } { 15 }$ 
B)  $\frac { 77 } { 225 }$ 
C)  $\frac { 19 } { 75 }$ 
D)  $\frac { 19 } { 5 }$

Question 2

Find the indicated vector.
-Let $\mathbf { u } = \langle 4 , - 6 \rangle$. Find $- 6 \mathbf { u }$.

Accepted Answer

A)  $\langle 24 , - 36 \rangle$ 
B)  $\langle 24,36 \rangle$ 
C)  $\langle - 24 , - 36 \rangle$ 
D)  $\langle - 24,36 \rangle$ 
A)  $\langle 24 , - 36 \rangle$ 
B)  $\langle 24,36 \rangle$ 
C)  $\langle - 24 , - 36 \rangle$ 
D)  $\langle - 24,36 \rangle$

Question 3

Solve the problem.
-Find the perimeter of the triangle with vertices A(2, 2, 6), B(2, -2, 6), and C(3, 5, 7).

Accepted Answer

A)  $4 \sqrt { 10 } + \sqrt { 51 } + \sqrt { 179 }$ 
B)  $0 + \sqrt { 51 } + \sqrt { 51 }$ 
C)  $4 + \sqrt { 51 } + \sqrt { 11 }$ 
D)  $4 \sqrt { 2 } + \sqrt { 51 } + \sqrt { 35 }$ 
A)  $4 \sqrt { 10 } + \sqrt { 51 } + \sqrt { 179 }$ 
B)  $0 + \sqrt { 51 } + \sqrt { 51 }$ 
C)  $4 + \sqrt { 51 } + \sqrt { 11 }$ 
D)  $4 \sqrt { 2 } + \sqrt { 51 } + \sqrt { 35 }$

Question 4

Describe the given set of points with a single equation or with a pair of equations.
-The plane through the point (1, -4, -5) and parallel to the yz-plane

Accepted Answer

A)  x = 1 
B)  1 + y + z = 0 
C)  -4y - 5z = 0 and x = 1 
D)  y = -4 and z = -5 
A)  x = 1 
B)  1 + y + z = 0 
C)  -4y - 5z = 0 and x = 1 
D)  y = -4 and z = -5

Question 5

Express the vector as a product of its length and direction.
-5i + 10j + 10k

Accepted Answer

A)  $15 \left( \frac { 1 } { 45 } \mathrm { i } + \frac { 2 } { 45 } \mathrm { j } + \frac { 2 } { 45 } \mathrm { k } \right)$ 
B)  $15 ( 5 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } )$ 
C)  $15 ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$ 
D)  $15 \left( \frac { 1 } { 3 } \mathrm { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 2 } { 3 } \mathrm { k } \right)$ 
A)  $15 \left( \frac { 1 } { 45 } \mathrm { i } + \frac { 2 } { 45 } \mathrm { j } + \frac { 2 } { 45 } \mathrm { k } \right)$ 
B)  $15 ( 5 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } )$ 
C)  $15 ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$ 
D)  $15 \left( \frac { 1 } { 3 } \mathrm { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 2 } { 3 } \mathrm { k } \right)$

Question 6

Solve the problem.
-A bird flies from its nest $6 \mathrm {~km}$ in the direction $11 ^ { \circ }$ north of east, where it stops to rest on a tree. It then flies $9 \mathrm {~km}$ in the direction $38 ^ { \circ }$ 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 tree located?

Accepted Answer

A)  (5.890, 1.145) 
B)  (0.02655, -6.000) 
C)  (1.145, 5.890) 
D)  (0.9816, 0.1908) 
A)  (5.890, 1.145) 
B)  (0.02655, -6.000) 
C)  (1.145, 5.890) 
D)  (0.9816, 0.1908)

Question 7

Find the length and direction (when defined) of u × v.
-u = -3i + 8i - 6k, v = 0

Accepted Answer

A)  $0 ; - 3 \mathbf { i } + 8 \mathbf { i } - 6 \mathbf { k }$ 
B)  $0 ; \frac { 1 } { \sqrt { 109 } } ( - 3 \mathbf { i } + 8 \mathbf { i } - 6 \mathbf { k } )$ 
C)  0; no direction 
D)  $\sqrt { 109 } ; \frac { 1 } { \sqrt { 109 } } ( - 3 \mathbf { i } + 8 \mathbf { i } - 6 \mathbf { k } )$ 
A)  $0 ; - 3 \mathbf { i } + 8 \mathbf { i } - 6 \mathbf { k }$ 
B)  $0 ; \frac { 1 } { \sqrt { 109 } } ( - 3 \mathbf { i } + 8 \mathbf { i } - 6 \mathbf { k } )$ 
C)  0; no direction 
D)  $\sqrt { 109 } ; \frac { 1 } { \sqrt { 109 } } ( - 3 \mathbf { i } + 8 \mathbf { i } - 6 \mathbf { k } )$

Question 8

Match the equation with the surface it defines.
-$$\frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$$

Accepted Answer

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

Question 9

Describe the given set of points with a single equation or with a pair of equations.
-The circle of radius 6 centered at the point (6, -4, 36) and lying in a plane parallel to the xy-plane

Accepted Answer

A)  $\left( \frac { x } { 6 } \right) ^ { 2 } + \left( \frac { y } { - 4 } \right) ^ { 2 } + 1 = 36$ 
B)  $( x - 6 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 36$ and $z = 36$ 
C)  $( x - 6 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 36$ and $x + y = 2$ 
D)  $\left( \frac { x } { 6 } \right) ^ { 2 } + \left( \frac { y } { - 4 } \right) ^ { 2 } = 36$ and $z = 36$ 
A)  $\left( \frac { x } { 6 } \right) ^ { 2 } + \left( \frac { y } { - 4 } \right) ^ { 2 } + 1 = 36$ 
B)  $( x - 6 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 36$ and $z = 36$ 
C)  $( x - 6 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 36$ and $x + y = 2$ 
D)  $\left( \frac { x } { 6 } \right) ^ { 2 } + \left( \frac { y } { - 4 } \right) ^ { 2 } = 36$ and $z = 36$

Question 10

Provide an appropriate response.
-Show that A = au + bv is orthogonal to B = bu - av, where u and v are orthogonal unit vectors.

Accepted Answer

@#LAT-DLM&
Since the dot produ

Question 11

Find the acute angle, in radians, between the lines.
-7x + 8y = -2 and 9x + 4y = -5

Accepted Answer

A)  0.9074 radians 
B)  1.137 radians 
C)  0.4337 radians 
D)  2.005 radians 
A)  0.9074 radians 
B)  1.137 radians 
C)  0.4337 radians 
D)  2.005 radians

Question 12

Calculate the requested distance.
-The distance from the point S(-6, 1, -4) to the line x = -2 + 2t, y = -1 + 6t, z = 6 + 9t

Accepted Answer

A)  $\frac { 2 \sqrt { 1781 } } { 11 }$ 
B)  $\frac { 7124 } { 121 }$ 
C)  $\frac { 7124 } { 11 }$ 
D)  $\frac { 2 \sqrt { 1781 } } { 121 }$ 
A)  $\frac { 2 \sqrt { 1781 } } { 11 }$ 
B)  $\frac { 7124 } { 121 }$ 
C)  $\frac { 7124 } { 11 }$ 
D)  $\frac { 2 \sqrt { 1781 } } { 121 }$

Question 13

Determine whether the following is always true or not always true. Given reasons for your answers.
-c(u · v) = cu ·cv (any number c)

Accepted Answer

The answer of Determine whether the following is always true...

Question 14

Express the vector as a product of its length and direction.
-$\frac { 1 } { \sqrt { 6 } } \mathrm { i } + \frac { 1 } { \sqrt { 6 } } \mathrm { j } - \frac { 1 } { \sqrt { 6 } } \mathbf { k }$

Accepted Answer

A)  $\frac { 1 } { 3 } \left( \frac { 3 } { \sqrt { 6 } } \mathbf { i } + \frac { 3 } { \sqrt { 6 } } \mathbf { j } - \frac { 3 } { \sqrt { 6 } } \mathbf { k } \right)$ 
B)  $\sqrt { \frac { 1 } { 2 } } \left( \frac { 1 } { \sqrt { 3 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } - \frac { 1 } { \sqrt { 3 } } \mathbf { k } \right)$ 
C)  $\sqrt { \frac { 1 } { 2 } } \left( \frac { 1 } { \sqrt { 6 } } \mathrm { i } + \frac { 1 } { \sqrt { 6 } } \mathrm { j } - \frac { 1 } { \sqrt { 6 } } \mathbf { k } \right)$ 
D)  $\frac { 1 } { 6 } \left( \frac { 6 } { \sqrt { 6 } } \mathbf { i } + \frac { 6 } { \sqrt { 6 } } \mathbf { j } - \frac { 6 } { \sqrt { 6 } } \mathbf { k } \right)$ 
A)  $\frac { 1 } { 3 } \left( \frac { 3 } { \sqrt { 6 } } \mathbf { i } + \frac { 3 } { \sqrt { 6 } } \mathbf { j } - \frac { 3 } { \sqrt { 6 } } \mathbf { k } \right)$ 
B)  $\sqrt { \frac { 1 } { 2 } } \left( \frac { 1 } { \sqrt { 3 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } - \frac { 1 } { \sqrt { 3 } } \mathbf { k } \right)$ 
C)  $\sqrt { \frac { 1 } { 2 } } \left( \frac { 1 } { \sqrt { 6 } } \mathrm { i } + \frac { 1 } { \sqrt { 6 } } \mathrm { j } - \frac { 1 } { \sqrt { 6 } } \mathbf { k } \right)$ 
D)  $\frac { 1 } { 6 } \left( \frac { 6 } { \sqrt { 6 } } \mathbf { i } + \frac { 6 } { \sqrt { 6 } } \mathbf { j } - \frac { 6 } { \sqrt { 6 } } \mathbf { k } \right)$

Question 15

Match the equation with the surface it defines.
-$$\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 36 } = \frac { z } { 5 }$$

Accepted Answer

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

Question 16

Solve the problem.
-A bullet is fired with a muzzle velocity of $1251 \mathrm { ft } / \mathrm { sec }$ from a gun aimed at an angle of $29 ^ { \circ }$ above the horizontal. Find the vertical component of the velocity.

Accepted Answer

A)  606.5 ft/sec 
B)  693.4 ft/sec 
C)  -935.8 ft/sec 
D)  1094 ft/sec 
A)  606.5 ft/sec 
B)  693.4 ft/sec 
C)  -935.8 ft/sec 
D)  1094 ft/sec

Question 17

Express the vector as a product of its length and direction.
-$$- 2 \mathbf { i } - 8 \mathbf { j } + \frac { 8 } { 3 } \mathbf { k }$$

Accepted Answer

A)  $\frac { 26 } { 3 } ( - \mathbf { i } - \mathrm { j } + \mathbf { k } )$ 
B)  $\frac { 26 } { 3 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } \right)$ 
C)  $\frac { 676 } { 9 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } \right)$ 
D)  $\frac { 3 } { 26 }$ 
A)  $\frac { 26 } { 3 } ( - \mathbf { i } - \mathrm { j } + \mathbf { k } )$ 
B)  $\frac { 26 } { 3 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } \right)$ 
C)  $\frac { 676 } { 9 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } \right)$ 
D)  $\frac { 3 } { 26 }$

Question 18

Sketch the given surface.
-$x^{2}+y^{2}=9$

Accepted Answer

The answer of Sketch the given surface.
-$x^{2}+y^{2}=9$
 ...

Question 19

Calculate the requested distance.
-The distance from the point S(7, 6, -8) to the line x = -2 + 4t, y = -9 + 3t, z = 3 + 12t

Accepted Answer

A)  $\frac { \sqrt { 69,562 } } { 13 }$ 
B)  $\frac { 69562 } { 169 }$ 
C)  $\frac { 69562 } { 13 }$ 
D)  $\frac { \sqrt { 69,562 } } { 169 }$ 
A)  $\frac { \sqrt { 69,562 } } { 13 }$ 
B)  $\frac { 69562 } { 169 }$ 
C)  $\frac { 69562 } { 13 }$ 
D)  $\frac { \sqrt { 69,562 } } { 169 }$

Question 20

Identify the type of surface represented by the given equation.
-$$\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 7 } + \frac { z ^ { 2 } } { 6 } = 1$$

Accepted Answer

A)  Ellipsoid 
B)  Paraboloid 
C)  Sphere 
D)  Elliptical cone 
A)  Ellipsoid 
B)  Paraboloid 
C)  Sphere 
D)  Elliptical cone

Calculate the requested distance. -The distance from the point S(1, -7, -9) to the plane 10x + 11y + 2z = -8

Find the indicated vector. -Let $\mathbf { u } = \langle 4 , - 6 \rangle$ . Find $- 6 \mathbf { u }$ .

Solve the problem. -Find the perimeter of the triangle with vertices A(2, 2, 6), B(2, -2, 6), and C(3, 5, 7).

Describe the given set of points with a single equation or with a pair of equations. -The plane through the point (1, -4, -5) and parallel to the yz-plane

Express the vector as a product of its length and direction. -5i + 10j + 10k

Find the length and direction (when defined) of u × v. -u = -3i + 8i - 6k, v = 0

Match the equation with the surface it defines. - $\frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$ $Match the equation with the surface it defines. - \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$

Describe the given set of points with a single equation or with a pair of equations. -The circle of radius 6 centered at the point (6, -4, 36) and lying in a plane parallel to the xy-plane

Provide an appropriate response. -Show that A = au + bv is orthogonal to B = bu - av, where u and v are orthogonal unit vectors.

Find the acute angle, in radians, between the lines. -7x + 8y = -2 and 9x + 4y = -5

Calculate the requested distance. -The distance from the point S(-6, 1, -4) to the line x = -2 + 2t, y = -1 + 6t, z = 6 + 9t

Determine whether the following is always true or not always true. Given reasons for your answers. -c(u · v) = cu ·cv (any number c)

Express the vector as a product of its length and direction. - $\frac { 1 } { \sqrt { 6 } } \mathrm { i } + \frac { 1 } { \sqrt { 6 } } \mathrm { j } - \frac { 1 } { \sqrt { 6 } } \mathbf { k }$

Match the equation with the surface it defines. - $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 36 } = \frac { z } { 5 }$ $Match the equation with the surface it defines. - \frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 36 } = \frac { z } { 5 }$

Solve the problem. -A bullet is fired with a muzzle velocity of $1251 \mathrm { ft } / \mathrm { sec }$ from a gun aimed at an angle of $29 ^ { \circ }$ above the horizontal. Find the vertical component of the velocity.

Express the vector as a product of its length and direction. - $- 2 \mathbf { i } - 8 \mathbf { j } + \frac { 8 } { 3 } \mathbf { k }$

Sketch the given surface. - $x^{2}+y^{2}=9$ $Sketch the given surface. - x^{2}+y^{2}=9$

Calculate the requested distance. -The distance from the point S(7, 6, -8) to the line x = -2 + 4t, y = -9 + 3t, z = 3 + 12t

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 7 } + \frac { z ^ { 2 } } { 6 } = 1$

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Exam 13: Vectors and the Geometry of Space