Question 1

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$.

Accepted Answer

A)  $\frac { \pm ( 2 \mathbf { i } - 8 \mathbf { j } ) } { 65 }$ 
B)  $\frac { \pm ( 2 \mathbf { i } + 8 \mathbf { j } ) } { 65 }$ 
C)  $\frac { \pm ( \mathbf { i } + 8 \mathbf { j } ) } { \sqrt { 65 } }$ 
D)  $\frac { \pm ( \mathbf { i } + \mathbf { j } ) } { \sqrt { 65 } }$ 
E)  $\frac { \pm ( i - 8 j ) } { 65 }$ 
A)  $\frac { \pm ( 2 \mathbf { i } - 8 \mathbf { j } ) } { 65 }$ 
B)  $\frac { \pm ( 2 \mathbf { i } + 8 \mathbf { j } ) } { 65 }$ 
C)  $\frac { \pm ( \mathbf { i } + 8 \mathbf { j } ) } { \sqrt { 65 } }$ 
D)  $\frac { \pm ( \mathbf { i } + \mathbf { j } ) } { \sqrt { 65 } }$ 
E)  $\frac { \pm ( i - 8 j ) } { 65 }$

Question 2

Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N $60 ^ { \circ } \mathrm { W }$ at a speed of 80 $\mathrm { km } / \mathrm { h }$. (This means that the direction from which the wind blows is $60 ^ { \circ }$ west of the northerly direction.) A pilot is steering a plane in the direction $\mathrm { N } \mathrm {} 45 ^ { \circ } \mathrm { E }$ at an airspeed (speed in still air) of $250 \mathrm {~km} / \mathrm { h }$. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth.

Accepted Answer

The answer of Velocities have both direction and magnitude and...

Question 3

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 4

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 5

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 6

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 7

Reduce the equation to one of the standard forms. $$z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$$

Accepted Answer

A)  $x ^ { 2 } + \frac { y ^ { 2 } } { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
B)  $( x - 4 ) ^ { 2 } + \frac { y ^ { 2 } } { \left( \frac { 1 } { 4 } \right) ^ { 2 } } + z ^ { 2 } = 1$ 
C)  $x ^ { 2 } + y ^ { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } - z ^ { 2 } = 1$ 
E)  $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 6 } - \frac { z ^ { 2 } } { 24 } = 1$ 
A)  $x ^ { 2 } + \frac { y ^ { 2 } } { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
B)  $( x - 4 ) ^ { 2 } + \frac { y ^ { 2 } } { \left( \frac { 1 } { 4 } \right) ^ { 2 } } + z ^ { 2 } = 1$ 
C)  $x ^ { 2 } + y ^ { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } - z ^ { 2 } = 1$ 
E)  $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 6 } - \frac { z ^ { 2 } } { 24 } = 1$

Question 8

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

A)  $| \mathbf { u } \times \mathbf { v } | = 5$ 
B)  $| \mathbf { u } \times \mathbf { v } | = 7$ 
C)  $| \mathbf { u } \times \mathbf { v } | = 35$ 
D)  $| \mathbf { u } \times \mathbf { v } | = 157$ 
E)  $| \mathbf { u } \times \mathbf { v } | = 1$ 
A)  $| \mathbf { u } \times \mathbf { v } | = 5$ 
B)  $| \mathbf { u } \times \mathbf { v } | = 7$ 
C)  $| \mathbf { u } \times \mathbf { v } | = 35$ 
D)  $| \mathbf { u } \times \mathbf { v } | = 157$ 
E)  $| \mathbf { u } \times \mathbf { v } | = 1$

Question 9

Find the midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

Accepted Answer

A)  $( 5 , - 3 , - 2 )$ 
B)  $( - 1 , - 3 , - 1 )$ 
C)  $( - 1,1 , - 1 )$ 
D)  $( - 1,1 , - 2 )$ 
A)  $( 5 , - 3 , - 2 )$ 
B)  $( - 1 , - 3 , - 1 )$ 
C)  $( - 1,1 , - 1 )$ 
D)  $( - 1,1 , - 2 )$

Question 10

Suppose you start at the origin, move along the x-axis a distance of units in the positive direction, and then move downward a distance of 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 11

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 12

Find the volume of the parallelepiped with adjacent edges $P Q , P R$, and $P S$.
$$P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$$

Accepted Answer

The answer of Find the volume of the parallelepiped with...

Question 13

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 14

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

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)  $\{ - 7 , - 23,25 \}$

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

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

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.

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

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?

Reduce the equation to one of the standard forms. $z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$

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 midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

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

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 volume of the parallelepiped with adjacent edges $P Q , P R$ , and $P S$ . $P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$

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

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

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Vector Calculus

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Exam 12: Vectors and the Geometry of Space

Find the unit vectors that are parallel to the tangent line to the curve y=2x2y = 2 x ^ { 2 }y=2x2 at the point (2,8)( 2,8 )(2,8) .

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)

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.

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

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?

Reduce the equation to one of the standard forms. z2=12x2+4y2−24z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24z2=12x2+4y2−24

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 midpoint of the line segment joining the given points. (4,−2,−3)( 4 , - 2 , - 3 )(4,−2,−3) and (6,−4,−1)( 6 , - 4 , - 1 )(6,−4,−1)

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

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 volume of the parallelepiped with adjacent edges PQ,PRP Q , P RPQ,PR , and PSP SPS . P(2,3,4),Q(4,5,4),R(4,3,5),S(4,3,4)P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )P(2,3,4),Q(4,5,4),R(4,3,5),S(4,3,4)

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 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 } = \{ - 5,2 , - 2 \}a=⟨7,5,1},b={−5,2,−2}

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

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

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

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.

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

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?

Reduce the equation to one of the standard forms. $z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$

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 midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

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 volume of the parallelepiped with adjacent edges $P Q , P R$ , and $P S$ . $P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$

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