Question 1

$$\text { Which of the given lines is parallel to the line } x = 8 + t , y = t , z = - 7 - 10 t \text { ? }$$

Accepted Answer

The answer of \[\text { Which of the given lines...

Question 2

$$\text { Find an equation of the set of all points equidistant from the points } A ( 3 , - 4 , - 5 ) \text { and } B ( - 6,4 , - 12 ) \text {. }$$

Accepted Answer

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

Question 3

Classify the surface. Select the correct answer.
$$22 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$$

Accepted Answer

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

Question 4

$$\text { Find the unit vectors that are parallel to the tangent line to the curve } y = 2 x ^ { 2 } \text { at the point } ( 4 ) \text {. }$$

Accepted Answer

The answer of \[\text { Find the unit vectors that...

Question 5

Draw a rectangular box with the origin and $( 3,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Accepted Answer

The answer of Draw a rectangular box with the origin...

Question 6

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$.

Accepted Answer

The answer of Let \(\mathbf { v } = 7...

Question 7

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

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 8

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 $20 \mathrm { mi } / \mathrm { h }$. 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 9

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 10

a. Find an equation of the sphere that passes through the point $( 6 , - 5,3 )$ and has center $( - 3,5,3 )$.
b. Find the curve in which this sphere intersects the $x y$-plane.

Accepted Answer

The answer of a. Find an equation of the sphere...

Question 11

Find the values of $x$ such that the vectors $( 2 x , x , 5 )$ and $( 4 , x , 3 )$ are orthogonal. Select the correct answer.

Accepted Answer

A)  $x = - 5 , x = - 3$ 
B)  $x = 5 , x = - 5$ 
C)  $x = - 5 , x = 3$ 
D)  $x = 3 , x = - 3$ 
E)  $x = 5 , x = 3$ 
A)  $x = - 5 , x = - 3$ 
B)  $x = 5 , x = - 5$ 
C)  $x = - 5 , x = 3$ 
D)  $x = 3 , x = - 3$ 
E)  $x = 5 , x = 3$

Question 12

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$, is perpendicular to the plane $x + y - 2 z = 2$.

Accepted Answer

The answer of Find an equation of the plane that...

Question 13

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

Accepted Answer

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

Question 14

Find the distance (correct to two decimal places) between the given parallel planes. Select the correct answer.
$6 x + 7 y - 2 z = 12$ and $12 x + 14 y - 4 z = 30$

Accepted Answer

A)  $0.32$ 
B)  $2.32$ 
C)  $4.32$ 
D)  $6.32$ 
E)  $9.32$ 
A)  $0.32$ 
B)  $2.32$ 
C)  $4.32$ 
D)  $6.32$ 
E)  $9.32$

Question 15

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 16

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 $( 8,6,11 )$ along a straight line. The distance is measured in meters and the force in newtons.
Select the correct answer.

Accepted Answer

A)  $97 \mathrm {~J}$ 
B)  $77 \mathrm {~J}$ 
C)  $107 \mathrm {~J}$ 
D)  $87 \mathrm {~J}$ 
E)  $117 \mathrm {~J}$ 
A)  $97 \mathrm {~J}$ 
B)  $77 \mathrm {~J}$ 
C)  $107 \mathrm {~J}$ 
D)  $87 \mathrm {~J}$ 
E)  $117 \mathrm {~J}$

Question 17

Find the distance (correct to two decimal places) between the given parallel planes. $8 x + 7 y - 2 z = 12$ and $18 x + 20 y - 6 z = 50$
Select the correct answer.

Accepted Answer

A)  $4.51$ 
B)  $2.51$ 
C)  $6.51$ 
D)  $0.51$ 
E)  $9.51$ 
A)  $4.51$ 
B)  $2.51$ 
C)  $6.51$ 
D)  $0.51$ 
E)  $9.51$

Question 18

Find, correct to the nearest degree, the angle between the planes. Select the correct answer.
$$x + y - z = 1,3 x - 13 y + 6 z = 5$$

Accepted Answer

A)  $129 ^ { \circ }$ 
B)  $8 ^ { \circ }$ 
C)  $5 ^ { \circ }$ 
D)  $102 ^ { \circ }$ 
E)  none of these 
A)  $129 ^ { \circ }$ 
B)  $8 ^ { \circ }$ 
C)  $5 ^ { \circ }$ 
D)  $102 ^ { \circ }$ 
E)  none of these

Question 19

Find an equation for the surface obtained by rotating the parabola $y = 2 x ^ { 2 }$ about the $y$-axis.
Select the correct answer.

Accepted Answer

A)  $z ^ { 2 } - 2 x ^ { 2 } - y ^ { 2 } = 1$ 
B)  $2 z ^ { 2 } + 2 x ^ { 2 } = y$ 
C)  $z ^ { 2 } - 2 x ^ { 2 } = y$ 
D)  $z ^ { 2 } + 2 x ^ { 2 } + y ^ { 2 } = 1$ 
E)  $y ^ { 2 } + 2 x ^ { 2 } = z$ 
A)  $z ^ { 2 } - 2 x ^ { 2 } - y ^ { 2 } = 1$ 
B)  $2 z ^ { 2 } + 2 x ^ { 2 } = y$ 
C)  $z ^ { 2 } - 2 x ^ { 2 } = y$ 
D)  $z ^ { 2 } + 2 x ^ { 2 } + y ^ { 2 } = 1$ 
E)  $y ^ { 2 } + 2 x ^ { 2 } = z$

Question 20

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the $x$-axis. Find the equation of the ellipsoid.

Accepted Answer

The answer of An ellipsoid is created by rotating the...

$\text { Which of the given lines is parallel to the line } x = 8 + t , y = t , z = - 7 - 10 t \text { ? }$

$\text { Find an equation of the set of all points equidistant from the points } A ( 3 , - 4 , - 5 ) \text { and } B ( - 6,4 , - 12 ) \text {. }$

Classify the surface. Select the correct answer. $22 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$

$\text { Find the unit vectors that are parallel to the tangent line to the curve } y = 2 x ^ { 2 } \text { at the point } ( 4 ) \text {. }$

Draw a rectangular box with the origin and $( 3,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .

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

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 $20 \mathrm { mi } / \mathrm { h }$ . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

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.

a. Find an equation of the sphere that passes through the point $( 6 , - 5,3 )$ and has center $( - 3,5,3 )$ . b. Find the curve in which this sphere intersects the $x y$ -plane.

Find the values of $x$ such that the vectors $( 2 x , x , 5 )$ and $( 4 , x , 3 )$ are orthogonal. Select the correct answer.

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ , is perpendicular to the plane $x + y - 2 z = 2$ .

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

Find the distance (correct to two decimal places) between the given parallel planes. Select the correct answer. $6 x + 7 y - 2 z = 12$ and $12 x + 14 y - 4 z = 30$

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

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 $( 8,6,11 )$ along a straight line. The distance is measured in meters and the force in newtons. Select the correct answer.

Find the distance (correct to two decimal places) between the given parallel planes. $8 x + 7 y - 2 z = 12$ and $18 x + 20 y - 6 z = 50$ Select the correct answer.

Find, correct to the nearest degree, the angle between the planes. Select the correct answer. $x + y - z = 1,3 x - 13 y + 6 z = 5$

Find an equation for the surface obtained by rotating the parabola $y = 2 x ^ { 2 }$ about the $y$ -axis. Select the correct answer.

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the $x$ -axis. Find the equation of the ellipsoid.

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

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Partial Derivatives

Multiple Integrals

Vector Calculus

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Exam 12: Vectors and the Geometry of Space

Which of the given lines is parallel to the line x=8+t,y=t,z=−7−10t ? \text { Which of the given lines is parallel to the line } x = 8 + t , y = t , z = - 7 - 10 t \text { ? } Which of the given lines is parallel to the line x=8+t,y=t,z=−7−10t ?

Classify the surface. Select the correct answer. 22x2+y2−z2−2y+2z=022 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 022x2+y2−z2−2y+2z=0

Draw a rectangular box with the origin and (3,6,3)( 3,6,3 )(3,6,3) as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Let v=7j\mathbf { v } = 7 \mathbf { j }v=7j and let u\mathbf { u }u be a vector with length 5 that starts at the origin and rotates in the xyx yxy - plane. Find the maximum value of the length of the vector ∣u×v∣| \mathbf { u } \times \mathbf { v } |∣u×v∣ .

Find the center and the radius of the sphere that has the given equation. Select the correct answer. 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 20mi/h20 \mathrm { mi } / \mathrm { h }20mi/h . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

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.

a. Find an equation of the sphere that passes through the point (6,−5,3)( 6 , - 5,3 )(6,−5,3) and has center (−3,5,3)( - 3,5,3 )(−3,5,3) . b. Find the curve in which this sphere intersects the xyx yxy -plane.

Find the values of xxx such that the vectors (2x,x,5)( 2 x , x , 5 )(2x,x,5) and (4,x,3)( 4 , x , 3 )(4,x,3) are orthogonal. Select the correct answer.

Find an equation of the plane that passes through the line of intersection of the planes x−z=1x - z = 1x−z=1 and y+2z=3y + 2 z = 3y+2z=3 , is perpendicular to the plane x+y−2z=2x + y - 2 z = 2x+y−2z=2 .

Find parametric equations for the line through the point (4,5,5)( 4,5,5 )(4,5,5) 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 . Select the correct answer.

Find the distance (correct to two decimal places) between the given parallel planes. Select the correct answer. 6x+7y−2z=126 x + 7 y - 2 z = 126x+7y−2z=12 and 12x+14y−4z=3012 x + 14 y - 4 z = 3012x+14y−4z=30

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 distance (correct to two decimal places) between the given parallel planes. 8x+7y−2z=128 x + 7 y - 2 z = 128x+7y−2z=12 and 18x+20y−6z=5018 x + 20 y - 6 z = 5018x+20y−6z=50 Select the correct answer.

Find, correct to the nearest degree, the angle between the planes. Select the correct answer. x+y−z=1,3x−13y+6z=5x + y - z = 1,3 x - 13 y + 6 z = 5x+y−z=1,3x−13y+6z=5

Find an equation for the surface obtained by rotating the parabola y=2x2y = 2 x ^ { 2 }y=2x2 about the yyy -axis. Select the correct answer.

An ellipsoid is created by rotating the ellipse 4x2+y2=164 x ^ { 2 } + y ^ { 2 } = 164x2+y2=16 about the xxx -axis. Find the equation of the ellipsoid.

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

$\text { Which of the given lines is parallel to the line } x = 8 + t , y = t , z = - 7 - 10 t \text { ? }$

Classify the surface. Select the correct answer. $22 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$

Draw a rectangular box with the origin and $( 3,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .

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

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 $20 \mathrm { mi } / \mathrm { h }$ . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

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.

a. Find an equation of the sphere that passes through the point $( 6 , - 5,3 )$ and has center $( - 3,5,3 )$ . b. Find the curve in which this sphere intersects the $x y$ -plane.

Find the values of $x$ such that the vectors $( 2 x , x , 5 )$ and $( 4 , x , 3 )$ are orthogonal. Select the correct answer.

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ , is perpendicular to the plane $x + y - 2 z = 2$ .

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

Find the distance (correct to two decimal places) between the given parallel planes. Select the correct answer. $6 x + 7 y - 2 z = 12$ and $12 x + 14 y - 4 z = 30$

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

Find the distance (correct to two decimal places) between the given parallel planes. $8 x + 7 y - 2 z = 12$ and $18 x + 20 y - 6 z = 50$ Select the correct answer.

Find, correct to the nearest degree, the angle between the planes. Select the correct answer. $x + y - z = 1,3 x - 13 y + 6 z = 5$

Find an equation for the surface obtained by rotating the parabola $y = 2 x ^ { 2 }$ about the $y$ -axis. Select the correct answer.

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the $x$ -axis. Find the equation of the ellipsoid.