Question 1

Plot the given points in a three-dimensional coordinate system. $( 3,2,3 )$

Accepted Answer

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

Question 2

Find, correct to the nearest degree, the angle between the planes. $$x + y - z = 1,10 x - 15 y + 20 z = 5$$

Accepted Answer

A)  $122 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $22 ^ { \circ }$ 
D)  $102 ^ { \circ }$ 
E)  none of these 
A)  $122 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $22 ^ { \circ }$ 
D)  $102 ^ { \circ }$ 
E)  none of these

Question 3

Find the center and radius of the sphere. $$x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$$

Accepted Answer

A)  $C ( - 1 , - 12 , - 13 ) , r = 6 \sqrt { 2 }$ 
B)  $C ( - 1 , - 12 , - 13 ) , r = \sqrt { 2 }$ 
C)  $C ( 1 , - 12 , - 13 ) , r = 6$ 
D)  $C ( - 1 , - 12 , - 13 ) , r = 6$ 
E)  none of these 
A)  $C ( - 1 , - 12 , - 13 ) , r = 6 \sqrt { 2 }$ 
B)  $C ( - 1 , - 12 , - 13 ) , r = \sqrt { 2 }$ 
C)  $C ( 1 , - 12 , - 13 ) , r = 6$ 
D)  $C ( - 1 , - 12 , - 13 ) , r = 6$ 
E)  none of these

Question 4

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

Accepted Answer

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

Question 5

Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$.

Accepted Answer

A)  $x = 4 t + 2 , y = - 2 t + 4 , z = - 2 t + 5$ 
B)  $x = 4 t + 2 , y = - 2 t + 4 , z = 0$ 
C)  $x = 4 t , y = - 2 t , z = - 2 t + 5$ 
D)  $x = 4 t , y = - 2 t + 4 , z = - 2 t$ 
E)  $x = 4 t + 3 , y = - 2 t + 4 , z = - 2 t + 5$ 
A)  $x = 4 t + 2 , y = - 2 t + 4 , z = - 2 t + 5$ 
B)  $x = 4 t + 2 , y = - 2 t + 4 , z = 0$ 
C)  $x = 4 t , y = - 2 t , z = - 2 t + 5$ 
D)  $x = 4 t , y = - 2 t + 4 , z = - 2 t$ 
E)  $x = 4 t + 3 , y = - 2 t + 4 , z = - 2 t + 5$

Question 6

Determine whether the given points are collinear. $A ( 2,3,2 ) , B ( - 4,0,5 )$, and $C ( 4,4,1 )$

Accepted Answer

A)  Not collinear 
B)  Collinear 
A)  Not collinear 
B)  Collinear

Question 7

Find the cross product $\mathbf { a } \times \mathbf { b }$.
$$\mathbf { a } = \langle 7,5,1 \} , \mathbf { b } = \langle - 5,2 , - 2 \}$$

Accepted Answer

A)  $( - 8 , - 12 , - 19 )$ 
B)  $\{ - 12,39,9 \}$ 
C)  $\{ - 12,9,39 )$ 
D)  $\{ 39 , - 12,9 \}$ 
E)  $\{ - 7 , - 23,25 \}$ 
A)  $( - 8 , - 12 , - 19 )$ 
B)  $\{ - 12,39,9 \}$ 
C)  $\{ - 12,9,39 )$ 
D)  $\{ 39 , - 12,9 \}$ 
E)  $\{ - 7 , - 23,25 \}$

Question 8

Find the distance from $( - 10,2 , - 8 )$ to the $x y$-planes.

Accepted Answer

A)  8 
B)  16 
C)  10 
D)  2 
E)  4 
A)  8 
B)  16 
C)  10 
D)  2 
E)  4

Question 9

$\text { Find an equation of the plane through the origin and parallel to the plane } - 3 x + 6 y - 3 z = - 1$

Accepted Answer

The answer of \(\text { Find an equation of the...

Question 10

Find the center and the radius of the sphere that has the given equation. $$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$$

Accepted Answer

A)  $( 2 , - 4,0 ) , 2 \sqrt { 5 }$ 
B)  $( - 2,4,0 ) , 2 \sqrt { 5 }$ 
C)  $( - 2,4,0 ) , 20$ 
D)  $( 2 , - 4,0 ) , 20$ 
A)  $( 2 , - 4,0 ) , 2 \sqrt { 5 }$ 
B)  $( - 2,4,0 ) , 2 \sqrt { 5 }$ 
C)  $( - 2,4,0 ) , 20$ 
D)  $( 2 , - 4,0 ) , 20$

Question 11

Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Accepted Answer

A)  $3,2 , \sqrt { 5 }$, right 
B)  $1,2 , \sqrt { 5 }$, neither 
C)  $1,1 , \sqrt { 2 }$, both 
D)  $1,1 , \sqrt { 2 }$, isosceles 
A)  $3,2 , \sqrt { 5 }$, right 
B)  $1,2 , \sqrt { 5 }$, neither 
C)  $1,1 , \sqrt { 2 }$, both 
D)  $1,1 , \sqrt { 2 }$, isosceles

Question 12

A bicycle pedal is pushed by a foot with a force of $a - \mathrm { N }$ as shown. The shaft of the pedal is $12 \mathrm {~cm}$ long. Find the magnitude of the torque about P correct to two decimal places. $$\text { Let } a = 70 , b = 70 ^ { \circ } , c = 10 ^ { \circ } \text {. }$$

Accepted Answer

The answer of A bicycle pedal is pushed by a...

Question 13

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Accepted Answer

A)  $64 \mathrm {~J}$ 
B)  $74 \mathrm {~J}$ 
C)  $84 \mathrm {~J}$ 
D)  $94 \mathrm {~J}$ 
E)  $104 \mathrm {~J}$ 
A)  $64 \mathrm {~J}$ 
B)  $74 \mathrm {~J}$ 
C)  $84 \mathrm {~J}$ 
D)  $94 \mathrm {~J}$ 
E)  $104 \mathrm {~J}$

Question 14

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$. Round your answer to two decimal place.

Accepted Answer

A)  $4.79$ 
B)  $6.79$ 
C)  $4.29$ 
D)  $7.79$ 
E)  $5.79$ 
A)  $4.79$ 
B)  $6.79$ 
C)  $4.29$ 
D)  $7.79$ 
E)  $5.79$

Question 15

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$.

Accepted Answer

The answer of Find the length of the median of...

Question 16

Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$-axis is two times the distance from $P$ to the $y z$-plane.

Accepted Answer

A)  $2 x ^ { 2 } = y ^ { 2 } - 2 z ^ { 2 }$ 
B)  $4 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ 
C)  $2 x ^ { 2 } = y ^ { 2 } + 2 z$ 
D)  $x ^ { 2 } - y ^ { 2 } - z ^ { 2 } = 2$ 
E)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ 
A)  $2 x ^ { 2 } = y ^ { 2 } - 2 z ^ { 2 }$ 
B)  $4 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ 
C)  $2 x ^ { 2 } = y ^ { 2 } + 2 z$ 
D)  $x ^ { 2 } - y ^ { 2 } - z ^ { 2 } = 2$ 
E)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$

Question 17

$\text { Write inequalities to describe the solid upper hemisphere of the sphere of radius } 3 \text { centered at the origin. }$

Accepted Answer

The answer of \(\text { Write inequalities to describe the...

Question 18

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 5 \mathbf { j }$ and $\mathbf { b } = - 5 \mathbf { j } + 2 \mathbf { k }$.
I. $\mathbf { a } \cdot \mathbf { b }$
П. $\mathbf { a } \times \mathbf { b }$
II. The angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree)

Accepted Answer

The answer of Calculate the given quantities if \(\mathbf {...

Question 19

Find a nonzero vector orthogonal to the plane through the points $P , Q$, and $R$.
$$P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$$

Accepted Answer

A)  $\mathbf { i } - 7 \mathbf { j } + 8 \mathbf { k }$ 
B)  $- 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k }$ 
C)  $\mathbf { i } - 8 \mathbf { j } + 7 \mathbf { k }$ 
D)  $8 \mathbf { i } - 6 \mathbf { j } + 56 \mathbf { k }$ 
E)  None of these 
A)  $\mathbf { i } - 7 \mathbf { j } + 8 \mathbf { k }$ 
B)  $- 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k }$ 
C)  $\mathbf { i } - 8 \mathbf { j } + 7 \mathbf { k }$ 
D)  $8 \mathbf { i } - 6 \mathbf { j } + 56 \mathbf { k }$ 
E)  None of these

Question 20

Find the distance between the planes. $$5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 3 } \sqrt { 30 }$ 
C)  $\frac { 5 \sqrt { 30 } } { 6 }$ 
D)  $\frac { 5 } { 6 }$ 
E)  $\sqrt { 30 }$ 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 3 } \sqrt { 30 }$ 
C)  $\frac { 5 \sqrt { 30 } } { 6 }$ 
D)  $\frac { 5 } { 6 }$ 
E)  $\sqrt { 30 }$

Plot the given points in a three-dimensional coordinate system. $( 3,2,3 )$

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,10 x - 15 y + 20 z = 5$

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$ .

Determine whether the given points are collinear. $A ( 2,3,2 ) , B ( - 4,0,5 )$ , and $C ( 4,4,1 )$

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 7,5,1 \} , \mathbf { b } = \langle - 5,2 , - 2 \}$

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

$\text { Find an equation of the plane through the origin and parallel to the plane } - 3 x + 6 y - 3 z = - 1$

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$

Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ . Round your answer to two decimal place.

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

$\text { Write inequalities to describe the solid upper hemisphere of the sphere of radius } 3 \text { centered at the origin. }$

Find a nonzero vector orthogonal to the plane through the points $P , Q$ , and $R$ . $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Exam 12: Vectors and the Geometry of Space

Plot the given points in a three-dimensional coordinate system. (3,2,3)( 3,2,3 )(3,2,3)

Find, correct to the nearest degree, the angle between the planes. x+y−z=1,10x−15y+20z=5x + y - z = 1,10 x - 15 y + 20 z = 5x+y−z=1,10x−15y+20z=5

Find the center and radius of the sphere. x2−6x−26+y2+24y+z2−18z=0x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0x2−6x−26+y2+24y+z2−18z=0

Sketch the plane in a three-dimensional space represented by the equation. z=−1z = - 1z=−1

Find parametric equations for the line through the point (3,4,5)( 3,4,5 )(3,4,5) that is parallel to the plane x+y+z=−15x + y + z = - 15x+y+z=−15 and perpendicular to the line x=15+t,y=12−t,z=3tx = 15 + t , y = 12 - t , z = 3 tx=15+t,y=12−t,z=3t .

Determine whether the given points are collinear. A(2,3,2),B(−4,0,5)A ( 2,3,2 ) , B ( - 4,0,5 )A(2,3,2),B(−4,0,5) , and C(4,4,1)C ( 4,4,1 )C(4,4,1)

Find the cross product a×b\mathbf { a } \times \mathbf { b }a×b . a=⟨7,5,1},b=⟨−5,2,−2}\mathbf { a } = \langle 7,5,1 \} , \mathbf { b } = \langle - 5,2 , - 2 \}a=⟨7,5,1},b=⟨−5,2,−2}

Find the distance from (−10,2,−8)( - 10,2 , - 8 )(−10,2,−8) to the xyx yxy -planes.

Find an equation of the plane through the origin and parallel to the plane −3x+6y−3z=−1\text { Find an equation of the plane through the origin and parallel to the plane } - 3 x + 6 y - 3 z = - 1 Find an equation of the plane through the origin and parallel to the plane −3x+6y−3z=−1

Find the center and the radius of the sphere that has the given equation. x2+y2+z2−4x+8y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0x2+y2+z2−4x+8y=0

Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. A(−1,0,1),B(1,1,−1),C(1,1,1)A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )A(−1,0,1),B(1,1,−1),C(1,1,1)

Find the work done by a force F=i+5j+7k\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }F=i+5j+7k that moves an object from the point (3,0,5)( 3,0,5 )(3,0,5) to the point (7,5,10)( 7,5,10 )(7,5,10) along a straight line. The distance is measured in meters and the force in newtons.

Find the distance between the point (9,9,3)( 9,9,3 )(9,9,3) and the plane 15x−3y−8z=1015 x - 3 y - 8 z = 1015x−3y−8z=10 . Round your answer to two decimal place.

Find the length of the median of side ABA BAB of the triangle with vertices A(−8,6,−9),B(−6,−6,−5)A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )A(−8,6,−9),B(−6,−6,−5) and C(8,−6,2)C ( 8 , - 6,2 )C(8,−6,2) .

Find an equation for the surface consisting of all points PPP for which the distance from PPP to the xxx -axis is two times the distance from PPP to the yzy zyz -plane.

Find a nonzero vector orthogonal to the plane through the points P,QP , QP,Q , and RRR . P(1,0,0),Q(7,8,0),R(0,8,1)P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )P(1,0,0),Q(7,8,0),R(0,8,1)

Find the distance between the planes. 5x−2y+z−4=0,5x−2y+z+6=05 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 05x−2y+z−4=0,5x−2y+z+6=0

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Plot the given points in a three-dimensional coordinate system. $( 3,2,3 )$

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,10 x - 15 y + 20 z = 5$

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$ .

Determine whether the given points are collinear. $A ( 2,3,2 ) , B ( - 4,0,5 )$ , and $C ( 4,4,1 )$

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 7,5,1 \} , \mathbf { b } = \langle - 5,2 , - 2 \}$

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

$\text { Find an equation of the plane through the origin and parallel to the plane } - 3 x + 6 y - 3 z = - 1$

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$

Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ . Round your answer to two decimal place.

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

Find a nonzero vector orthogonal to the plane through the points $P , Q$ , and $R$ . $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$