Question 1

Find the distance between the given point Q and the given line.
-Q(-9, 1, 10); x = -5 + 3t, y = 1 - t, z = 7 + 4t

Accepted Answer

A)  6   
B)  5   
C)  5 
D)  6 
A)  6   
B)  5   
C)  5 
D)  6

Question 2

Express the vector in the form ai + bj.
-  if   is the point ( 2, -6) and   is the point ( 4, -9)

Accepted Answer

A)  v = -i + 3j 
B)  v = 2i - 3j 
C)  v = -2i + 3j 
D)  v = -2i - 3j 
A)  v = -i + 3j 
B)  v = 2i - 3j 
C)  v = -2i + 3j 
D)  v = -2i - 3j

Question 3

Find v ∙ u.
-v = 5i + 4j and u = 2i + 6j

Accepted Answer

A)  34 
B)  7i + 10j 
C)  -14 
D)  10i + 24j 
A)  34 
B)  7i + 10j 
C)  -14 
D)  10i + 24j

Question 4

Write the equation for the plane.
-The plane through the points P( 5, -3, -1) , Q( -3, 5, -33) and R( -1, 4, -32).

Accepted Answer

A)  7x + y + 3z = 7 
B)  7x + y + 3z = -7 
C)  3x + 7y + z = -7 
D)  3x + 7y + z = 7 
A)  7x + y + 3z = 7 
B)  7x + y + 3z = -7 
C)  3x + 7y + z = -7 
D)  3x + 7y + z = 7

Question 5

Write the equation for the plane.
-The plane through the point P( 2, 2, 7) and normal to n =

Accepted Answer

A)  2x + 2y - 7z = -16 
B)  4x + 3y - 2z = -16 
C)  -4x - 3y + 2z = 0 
D)  -2x - 2y + 7z = -16 
A)  2x + 2y - 7z = -16 
B)  4x + 3y - 2z = -16 
C)  -4x - 3y + 2z = 0 
D)  -2x - 2y + 7z = -16

Question 6

Find u.   
-v = 2i - 2j - 4k, u = 12i - 5k

Accepted Answer

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

Question 7

Find u.   
-v = k, u = 6i + 9j + 2k

Accepted Answer

A)  
  
B)  
  
C)  
  
D) 2k 
A)  
  
B)  
  
C)  
  
D) 2k

Question 8

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

Accepted Answer

Not always

Question 9

Solve the problem.
-A bullet is fired with a muzzle velocity of 1159 ft/sec from a gun aimed at an angle of 29° above the horizontal. Find the horizontal component of the velocity.

Accepted Answer

A)  1014 ft/sec 
B)  -867.0 ft/sec 
C)  561.9 ft/sec 
D)  642.4 ft/sec 
A)  1014 ft/sec 
B)  -867.0 ft/sec 
C)  561.9 ft/sec 
D)  642.4 ft/sec

Question 10

Find the indicated vector.
-Let u =   , v =   . Find   u +   v.

Accepted Answer

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

Question 11

Identify the type of surface represented by the given equation.
-  -   =

Accepted Answer

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

Question 12

Find u.   
-v = 3i - j + 3k, u = 11i + 2j + 10k

Accepted Answer

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

Question 13

Find the intersection.
--4x - 3y + 9z = 10, 6x - 8y + 2z = -2

Accepted Answer

A)  x = 66t + 86, y = 62t - 52, z = -50t 
B)  x = 3300t + 86, y = 3100t + 52, z = -50t 
C)  x = -66t - 86, y = -62t + -52, z = -50t 
D)  x = 66t -   , y = 62t -   , z = 50t 
A)  x = 66t + 86, y = 62t - 52, z = -50t 
B)  x = 3300t + 86, y = 3100t + 52, z = -50t 
C)  x = -66t - 86, y = -62t + -52, z = -50t 
D)  x = 66t -   , y = 62t -   , z = 50t

Question 14

Find the indicated vector.
-Let u =   . Find -8u.

Accepted Answer

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

Question 15

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

Accepted Answer

A)  ( 3 - 3   )i + 0j 
B)  -49 - 9   
C)  -49   i - 9j 
D)  -49   + 9 
A)  ( 3 - 3   )i + 0j 
B)  -49 - 9   
C)  -49   i - 9j 
D)  -49   + 9

Question 16

Identify the quadric surface by name. Find and describe the xy-, xz-, and yz-traces, when they exist.
--9   - 9   +   = 9

Accepted Answer

A)   Ellipsoid; xy-trace: -9xS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9 (circle); xz-trace: zS1U1P12 S1S1P0 - 9 xS1U1P12S1S1P0  = 9 (ellipse); yz-trace:  zS1U1P12S1S1P0 - 9 yS1U1P12S1S1P0 = 9 (ellipse) 
B)   Hyperboloid of two sheets; xy-trace: -9xS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9(circle); xz-trace: zS1U1P12 S1S1P0 - 9 xS1U1P12S1S1P0  = 9 (hyperbola); yz-trace:   zS1U1P12 S1S1P0 - 9 yS1U1P12S1S1P0  = 9 (hyperbola) 
C)   Hyperboloid of two sheets; xz-trace: zS1U1P12 S1S1P0 - 9 xS1U1P12S1S1P0  = 9 (hyperbola); yz-trace:  zS1U1P12 S1S1P0 - 9 yS1U1P12S1S1P0  = 9(hyperbola) 
D)   Hyperboloid of one sheet; xy-trace: -9xS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9 (circle); xz-trace:  zS1U1P12 S1S1P0 - 9 yS1U1P12S1S1P0  = 9 (hyperbola); yz-trace: zS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9 (hyperbola) 
A)   Ellipsoid; xy-trace: -9xS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9 (circle); xz-trace: zS1U1P12 S1S1P0 - 9 xS1U1P12S1S1P0  = 9 (ellipse); yz-trace:  zS1U1P12S1S1P0 - 9 yS1U1P12S1S1P0 = 9 (ellipse) 
B)   Hyperboloid of two sheets; xy-trace: -9xS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9(circle); xz-trace: zS1U1P12 S1S1P0 - 9 xS1U1P12S1S1P0  = 9 (hyperbola); yz-trace:   zS1U1P12 S1S1P0 - 9 yS1U1P12S1S1P0  = 9 (hyperbola) 
C)   Hyperboloid of two sheets; xz-trace: zS1U1P12 S1S1P0 - 9 xS1U1P12S1S1P0  = 9 (hyperbola); yz-trace:  zS1U1P12 S1S1P0 - 9 yS1U1P12S1S1P0  = 9(hyperbola) 
D)   Hyperboloid of one sheet; xy-trace: -9xS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9 (circle); xz-trace:  zS1U1P12 S1S1P0 - 9 yS1U1P12S1S1P0  = 9 (hyperbola); yz-trace: zS1U1P12S1S1P0  - 9yS1U1P12S1S1P0  = 9 (hyperbola)

Question 17

Solve the problem.
-How much work does it take to slide a box 6 meters along the ground by pulling it with a 158 N force at an angle of 15° from the horizontal?

Accepted Answer

A)  915.7 joules 
B)  -720.2 joules 
C)  152.6 joules 
D)  948 joules 
A)  915.7 joules 
B)  -720.2 joules 
C)  152.6 joules 
D)  948 joules

Question 18

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
-8x + 6y + 6z = -3 and 10x + 9y + 3z = -8

Accepted Answer

A)  1.446 rad 
B)  0.331 rad 
C)  1.521 rad 
D)  1.239 rad 
A)  1.446 rad 
B)  0.331 rad 
C)  1.521 rad 
D)  1.239 rad

Question 19

Find the triple scalar product (u x v) ∙ w of the given vectors.
-u = i + j + k; v = 8i + 5j + 2k; w = 9i + 6j + 4k

Accepted Answer

A)  -9 
B)  25 
C)  75 
D)  -3 
A)  -9 
B)  25 
C)  75 
D)  -3

Question 20

Write the equation for the plane.
-The plane passing through the point P0(-3, 1, 7) that is perpendicular to the line x = -1 + 6t, y = 2 - 5t,

Accepted Answer

A)  6x - 5y + 4z = 5 
B)  -3x + y + 7z = 68 
C)  4x + 9y + 2z = 11 
D)  -x + 2y + 3z = 26 
A)  6x - 5y + 4z = 5 
B)  -3x + y + 7z = 68 
C)  4x + 9y + 2z = 11 
D)  -x + 2y + 3z = 26

Find the distance between the given point Q and the given line. -Q(-9, 1, 10); x = -5 + 3t, y = 1 - t, z = 7 + 4t

Express the vector in the form ai + bj. - if is the point ( 2, -6) and is the point ( 4, -9)

Find v ∙ u. -v = 5i + 4j and u = 2i + 6j

Write the equation for the plane. -The plane through the points P( 5, -3, -1) , Q( -3, 5, -33) and R( -1, 4, -32).

Write the equation for the plane. -The plane through the point P( 2, 2, 7) and normal to n =

Find u. -v = 2i - 2j - 4k, u = 12i - 5k

Find u. -v = k, u = 6i + 9j + 2k

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

Solve the problem. -A bullet is fired with a muzzle velocity of 1159 ft/sec from a gun aimed at an angle of 29° above the horizontal. Find the horizontal component of the velocity.

Find the indicated vector. -Let u = , v = . Find u + v.

Identify the type of surface represented by the given equation. - - =

Find u. -v = 3i - j + 3k, u = 11i + 2j + 10k

Find the intersection. --4x - 3y + 9z = 10, 6x - 8y + 2z = -2

Find the indicated vector. -Let u = . Find -8u.

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

Identify the quadric surface by name. Find and describe the xy-, xz-, and yz-traces, when they exist. --9 - 9 + = 9

Solve the problem. -How much work does it take to slide a box 6 meters along the ground by pulling it with a 158 N force at an angle of 15° from the horizontal?

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. -8x + 6y + 6z = -3 and 10x + 9y + 3z = -8

Find the triple scalar product (u x v) ∙ w of the given vectors. -u = i + j + k; v = 8i + 5j + 2k; w = 9i + 6j + 4k

Write the equation for the plane. -The plane passing through the point P₀(-3, 1, 7) that is perpendicular to the line x = -1 + 6t, y = 2 - 5t,

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Exam 13: Vectors and the Geometry of Space