Question 1

Sketch the plane in a three-dimensional space represented by the equation.
$$z = 2$$

Accepted Answer

The answer of Sketch the plane in a three-dimensional space...

Question 2

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

The answer of Find a nonzero vector orthogonal to the...

Question 3

$$\text { Find parametric equations for the line through } ( - 9,6,7 ) \text { and parallel to the vector } ( 4,7 , - 4 \} \text {. }$$

Accepted Answer

The answer of \[\text { Find parametric equations for the...

Question 4

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

Question 5

A woman walks due west on the deck of a ship at $6 \mathrm { mi } / \mathrm { h }$. The ship is moving north at a speed of $21 \mathrm { mi } / \mathrm { h }$. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Accepted Answer

The answer of A woman walks due west on the...

Question 6

Classify the surface.
$$25 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$$
Select the correct answer.

Accepted Answer

A)  A hyperboloid of one sheet with center $( - 3,0,1 )$ and axis parallel to the $z$-axis. 
B)  A circular paraboloid with vertex $( 0,3,1 )$ and axis the $z$-axis. 
C)  A cone with axis parallel to the $z$-axis and vertex $( 0,1,1 )$. 
A)  A hyperboloid of one sheet with center $( - 3,0,1 )$ and axis parallel to the $z$-axis. 
B)  A circular paraboloid with vertex $( 0,3,1 )$ and axis the $z$-axis. 
C)  A cone with axis parallel to the $z$-axis and vertex $( 0,1,1 )$.

Question 7

Find the standard equation of the sphere with center $C$ and radius $r$.
$$C ( 4 , - 1,1 ) ; r = 6$$ .

Accepted Answer

The answer of Find the standard equation of the sphere...

Question 8

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

Question 9

$$\text { Find parametric equations for the line through } ( - 5,3,3 ) \text { and } ( 5,6,7 ) \text {. }$$

Accepted Answer

The answer of \[\text { Find parametric equations for the...

Question 10

Find the point at which the line given by the parametric equations below intersects the plane.
$$2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , \quad z = 7 t$$

Accepted Answer

The answer of Find the point at which the line...

Question 11

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 12

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

Accepted Answer

The answer of Find an equation for the surface consisting...

Question 13

Suppose you start at the origin, move along the $x$-axis a distance of 5 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

Accepted Answer

A)  $( 0,5,3 )$ 
B)  $( 5,3,0 )$ 
C)  $( 5 , - 3,0 )$ 
D)  $( 5,0 , - 3 )$ 
E)  $( 5,0,3 )$ 
A)  $( 0,5,3 )$ 
B)  $( 5,3,0 )$ 
C)  $( 5 , - 3,0 )$ 
D)  $( 5,0 , - 3 )$ 
E)  $( 5,0,3 )$

Question 14

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

Accepted Answer

The answer of Find the distance between the planes.
\[5 x...

Question 15

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.
Select the correct answer.

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 16

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)  $\sqrt { 30 }$

Question 17

Reduce the equation to one of the standard forms.
$$z ^ { 2 } = 15 x ^ { 2 } + 5 y ^ { 2 } - 30$$

Accepted Answer

The answer of Reduce the equation to one of the...

Question 18

Find the distance between the planes.
$$5 x - 2 y + z - 3 = 0,5 x - 2 y + z + 5 = 0$$

Accepted Answer

The answer of Find the distance between the planes.
\[5 x...

Question 19

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

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 20

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }$.

Accepted Answer

A)  $69.179$ 
B)  $73.179$ 
C)  $53.179$ 
D)  $75.179$ 
A)  $69.179$ 
B)  $73.179$ 
C)  $53.179$ 
D)  $75.179$

Sketch the plane in a three-dimensional space represented by the equation. $z = 2$

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

$\text { Find parametric equations for the line through } ( - 9,6,7 ) \text { and parallel to the vector } ( 4,7 , - 4 \} \text {. }$

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

A woman walks due west on the deck of a ship at $6 \mathrm { mi } / \mathrm { h }$ . The ship is moving north at a speed of $21 \mathrm { mi } / \mathrm { h }$ . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Classify the surface. $25 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$ Select the correct answer.

Find the standard equation of the sphere with center $C$ and radius $r$ . $C ( 4 , - 1,1 ) ; r = 6$ .

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

$\text { Find parametric equations for the line through } ( - 5,3,3 ) \text { and } ( 5,6,7 ) \text {. }$

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , \quad z = 7 t$

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

Suppose you start at the origin, move along the $x$ -axis a distance of 5 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

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

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. Select the correct answer.

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

Reduce the equation to one of the standard forms. $z ^ { 2 } = 15 x ^ { 2 } + 5 y ^ { 2 } - 30$

Find the distance between the planes. $5 x - 2 y + z - 3 = 0,5 x - 2 y + z + 5 = 0$

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

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }$ .

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

Sketch the plane in a three-dimensional space represented by the equation. z=2z = 2z=2

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 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

A woman walks due west on the deck of a ship at 6mi/h6 \mathrm { mi } / \mathrm { h }6mi/h . The ship is moving north at a speed of 21mi/h21 \mathrm { mi } / \mathrm { h }21mi/h . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Classify the surface. 25x2+y2−z2−2y+2z=025 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 025x2+y2−z2−2y+2z=0 Select the correct answer.

Find the standard equation of the sphere with center CCC and radius rrr . C(4,−1,1);r=6C ( 4 , - 1,1 ) ; r = 6C(4,−1,1);r=6 .

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 parametric equations for the line through (−5,3,3) and (5,6,7). \text { Find parametric equations for the line through } ( - 5,3,3 ) \text { and } ( 5,6,7 ) \text {. } Find parametric equations for the line through (−5,3,3) and (5,6,7).

Find the point at which the line given by the parametric equations below intersects the plane. 2x+4y−3z=−48;x=10+7t,y=−10,z=7t2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , \quad z = 7 t2x+4y−3z=−48;x=10+7t,y=−10,z=7t

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

Suppose you start at the origin, move along the xxx -axis a distance of 5 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

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

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. Select the correct answer.

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

Reduce the equation to one of the standard forms. z2=15x2+5y2−30z ^ { 2 } = 15 x ^ { 2 } + 5 y ^ { 2 } - 30z2=15x2+5y2−30

Find the distance between the planes. 5x−2y+z−3=0,5x−2y+z+5=05 x - 2 y + z - 3 = 0,5 x - 2 y + z + 5 = 05x−2y+z−3=0,5x−2y+z+5=0

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

Find ∣u×v∣| \mathbf { u } \times \mathbf { v } |∣u×v∣ correct to three decimal places where ∣u∣=18,∣v∣=3,∠θ=80∘| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }∣u∣=18,∣v∣=3,∠θ=80∘ .

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

Sketch the plane in a three-dimensional space represented by the equation. $z = 2$

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

A woman walks due west on the deck of a ship at $6 \mathrm { mi } / \mathrm { h }$ . The ship is moving north at a speed of $21 \mathrm { mi } / \mathrm { h }$ . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Classify the surface. $25 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$ Select the correct answer.

Find the standard equation of the sphere with center $C$ and radius $r$ . $C ( 4 , - 1,1 ) ; r = 6$ .

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

$\text { Find parametric equations for the line through } ( - 5,3,3 ) \text { and } ( 5,6,7 ) \text {. }$

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , \quad z = 7 t$

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

Suppose you start at the origin, move along the $x$ -axis a distance of 5 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

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

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. Select the correct answer.

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

Reduce the equation to one of the standard forms. $z ^ { 2 } = 15 x ^ { 2 } + 5 y ^ { 2 } - 30$

Find the distance between the planes. $5 x - 2 y + z - 3 = 0,5 x - 2 y + z + 5 = 0$

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

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }$ .