Question 1

Find a nonzero vector orthogonal to the plane through the points  P,QP , QP,Q , and  RRR . Select the correct answer.
 P(1,0,0) ,Q(5,4,0) ,R(0,4,1) P ( 1,0,0 )  , Q ( 5,4,0 )  , R ( 0,4,1 ) P(1,0,0) ,Q(5,4,0) ,R(0,4,1)

Accepted Answer

A)    i−4j+5k\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }i−4j+5k 
B)    −5i+j+4k- 5 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }−5i+j+4k 
C)    i−5j+4k i - 5 j + 4 ki−5j+4k 
D)    4i−4j+20k4 \mathbf { i } - 4 \mathbf { j } + 20 \mathbf { k }4i−4j+20k 
E)   None of these

Question 2

Find the cross product $\mathbf { a } \times \mathbf { b }$.
$$\mathbf { a } = \{ 5,5,3 \} , \mathbf { b } = \langle - 5,4 , - 4 \}$$

Accepted Answer

The answer of Find the cross product \(\mathbf { a...

Question 3

Calculate the angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree).
$$\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$$

Accepted Answer

The answer of Calculate the angle between \(\mathbf { a...

Question 4

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

Accepted Answer

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

Question 5

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)

Accepted Answer

A)    i−7j+8k\mathbf { i } - 7 \mathbf { j } + 8 \mathbf { k }i−7j+8k 
B)    −7i+j+8k- 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k }−7i+j+8k 
C)    i−8j+7k\mathbf { i } - 8 \mathbf { j } + 7 \mathbf { k }i−8j+7k 
D)    8i−6j+56k8 \mathbf { i } - 6 \mathbf { j } + 56 \mathbf { k }8i−6j+56k 
E)   None of these

Question 6

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 9 \mathbf { j }$ and $\mathbf { b } = - 9 \mathbf { j } + 8 \mathbf { k }$
I. $\mathbf { a } \cdot \mathbf { b }$
II. $\mathbf { a } \times \mathbf { b }$
III. 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 7

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

The answer of Find parametric equations for the line through...

Question 8

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)

Accepted Answer

A)   Not collinear
B)   Collinear

Question 9

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

Accepted Answer

The answer of \[\text { Find the distance from }...

Question 10

A bicycle pedal is pushed by a foot with a force of $a - \mathrm { N }$ as shown. The shaft of the pedal is $13 \mathrm {~cm}$ long.
Find the magnitude of the torque about $P$ correct to two decimal places.
Let $a = 64 , b = 70 ^ { \circ } , c = 10 ^ { \circ }$.

Accepted Answer

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

Question 11

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

Accepted Answer

A)  
B)  
C)  
D)

Question 12

Find an equation of the sphere with center  (7,−5,7) ( 7 , - 5,7 ) (7,−5,7)   that touches the  xyx yxy -plane.

Accepted Answer

A)    (x+7) 2+(y−5) 2+(z+7) 2=49( x + 7 )  ^ { 2 } + ( y - 5 )  ^ { 2 } + ( z + 7 )  ^ { 2 } = 49(x+7) 2+(y−5) 2+(z+7) 2=49 
B)    (x+7) 2+(y+5) 2+(z−7) 2=49( x + 7 )  ^ { 2 } + ( y + 5 )  ^ { 2 } + ( z - 7 )  ^ { 2 } = 49(x+7) 2+(y+5) 2+(z−7) 2=49 
C)    (x−7) 2+(y+5) 2+(z−7) 2=40( x - 7 )  ^ { 2 } + ( y + 5 )  ^ { 2 } + ( z - 7 )  ^ { 2 } = 40(x−7) 2+(y+5) 2+(z−7) 2=40 
D)    (x−7) 2+(y+5) 2+(z−7) 2=49( x - 7 )  ^ { 2 } + ( y + 5 )  ^ { 2 } + ( z - 7 )  ^ { 2 } = 49(x−7) 2+(y+5) 2+(z−7) 2=49 
E)    (x−7) 2+(y+5) 2+(z−7) 2=7( x - 7 )  ^ { 2 } + ( y + 5 )  ^ { 2 } + ( z - 7 )  ^ { 2 } = 7(x−7) 2+(y+5) 2+(z−7) 2=7

Question 13

Find, correct to the nearest degree, the angle between the planes. Select the correct answer.
 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

Accepted Answer

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

Question 14

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  (1,0,3) ( 1,0,3 ) (1,0,3)   to the point  (5,3,8) ( 5,3,8 ) (5,3,8)   along a straight line. The distance is measured in meters and the force in newtons.
Select the correct answer.

Accepted Answer

A)    94 J94 \mathrm {~J}94 J 
B)    74 J74 \mathrm {~J}74 J 
C)    84 J84 \mathrm {~J}84 J 
D)    64 J64 \mathrm {~J}64 J 
E)    54 J54 \mathrm {~J}54 J

Question 15

Find the point of intersection.
$$L _ { 1 } : \frac { x - 17 } { 3 } = \frac { y - 58 } { 8 } = \frac { z - 23 } { 2 } , L _ { 2 } : \frac { x - 49 } { 7 } = \frac { y - 26 } { 4 } = z - 15$$

Accepted Answer

The answer of Find the point of intersection.
\[L _ {...

Question 16

Find the center and the radius of the sphere that has the given equation.
$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 10 y = 0$$

Accepted Answer

The answer of Find the center and the radius of...

Question 17

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$.

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 4,8 , - 5 ) , B ( - 2 , - 8 , - 3 )$ and $C ( 12 , - 6,8 )$.

Accepted Answer

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

Question 20

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 1,0,3 )$ to the point $( 4,2,7 )$ along a straight line. The distance is measured in meters and the force in newtons.

Accepted Answer

The answer of Find the work done by a force...

Exam 12: Vectors and the Geometry of Space

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

Correct Answer
verified

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

Correct Answer
verified

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 10 y = 0$

Correct Answer
verified

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 1,0,3 )$ to the point $( 4,2,7 )$ along a straight line. The distance is measured in meters and the force in newtons.

Correct Answer
verified

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

Correct Answer
verified

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 1,0,3 )$ to the point $( 5,3,8 )$ along a straight line. The distance is measured in meters and the force in newtons. Select the correct answer.

Correct Answer
verified

Calculate the angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree). $\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

Correct Answer
verified

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

Correct Answer
verified

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \{ 5,5,3 \} , \mathbf { b } = \langle - 5,4 , - 4 \}$

Correct Answer
verified

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 $x = 15 + t , y = 12 - t , z = 3 t$ .

Correct Answer
verified

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

Correct Answer
verified

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 4,8 , - 5 ) , B ( - 2 , - 8 , - 3 )$ and $C ( 12 , - 6,8 )$ .

Correct Answer
verified

Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$ -plane.

Correct Answer
verified

Correct Answer
verified

Find a nonzero vector orthogonal to the plane through the points $P , Q$ , and $R$ . Select the correct answer. $P ( 1,0,0 ) , Q ( 5,4,0 ) , R ( 0,4,1 )$

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Find the point of intersection. $L _ { 1 } : \frac { x - 17 } { 3 } = \frac { y - 58 } { 8 } = \frac { z - 23 } { 2 } , L _ { 2 } : \frac { x - 49 } { 7 } = \frac { y - 26 } { 4 } = z - 15$

Correct Answer
verified

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

Correct Answer
verified

Exam 12: Vectors and the Geometry of Space

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)

Correct AnswerverifiedShow Answer

Find an equation for the surface consisting of all points that are equidistant from the point (0,−2,0)( 0 , - 2,0 )(0,−2,0) and the plane y=2y = 2y=2 .

Correct AnswerverifiedShow Answer

Find the center and the radius of the sphere that has the given equation. x2+y2+z2−6x+10y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 10 y = 0x2+y2+z2−6x+10y=0

Correct AnswerverifiedShow Answer

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 (1,0,3)( 1,0,3 )(1,0,3) to the point (4,2,7)( 4,2,7 )(4,2,7) along a straight line. The distance is measured in meters and the force in newtons.

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Calculate the angle between a\mathbf { a }a and b\mathbf { b }b (correct to the nearest degree). a=3i−2j,b=−7j+k\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }a=3i−2j,b=−7j+k

Correct AnswerverifiedShow Answer

Find the distance from (−10,2,−8) to the xy-planes. \text { Find the distance from } ( - 10,2 , - 8 ) \text { to the } x y \text {-planes. } Find the distance from (−10,2,−8) to the xy-planes.

Correct AnswerverifiedShow Answer

Find the cross product a×b\mathbf { a } \times \mathbf { b }a×b . a={5,5,3},b=⟨−5,4,−4}\mathbf { a } = \{ 5,5,3 \} , \mathbf { b } = \langle - 5,4 , - 4 \}a={5,5,3},b=⟨−5,4,−4}

Correct AnswerverifiedShow Answer

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 .

Correct AnswerverifiedShow Answer

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)

Correct AnswerverifiedShow Answer

Find the length of the median of side ABA BAB of the triangle with vertices A(−4,8,−5),B(−2,−8,−3)A ( - 4,8 , - 5 ) , B ( - 2 , - 8 , - 3 )A(−4,8,−5),B(−2,−8,−3) and C(12,−6,8)C ( 12 , - 6,8 )C(12,−6,8) .

Correct AnswerverifiedShow Answer

Find an equation of the sphere with center (7,−5,7) ( 7 , - 5,7 ) (7,−5,7) that touches the xyx yxy -plane.

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find a nonzero vector orthogonal to the plane through the points P,QP , QP,Q , and RRR . Select the correct answer. P(1,0,0) ,Q(5,4,0) ,R(0,4,1) P ( 1,0,0 ) , Q ( 5,4,0 ) , R ( 0,4,1 ) P(1,0,0) ,Q(5,4,0) ,R(0,4,1)

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find, correct to the nearest degree, the angle between the planes. Select the correct answer. 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

Correct AnswerverifiedShow Answer

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

Correct Answer
verified

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

Correct Answer
verified

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 10 y = 0$

Correct Answer
verified

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 1,0,3 )$ to the point $( 4,2,7 )$ along a straight line. The distance is measured in meters and the force in newtons.

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Calculate the angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree). $\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

Correct Answer
verified

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

Correct Answer
verified

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \{ 5,5,3 \} , \mathbf { b } = \langle - 5,4 , - 4 \}$

Correct Answer
verified

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 $x = 15 + t , y = 12 - t , z = 3 t$ .

Correct Answer
verified

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

Correct Answer
verified

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 4,8 , - 5 ) , B ( - 2 , - 8 , - 3 )$ and $C ( 12 , - 6,8 )$ .

Correct Answer
verified

Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$ -plane.

Correct Answer
verified

Correct Answer
verified

Find a nonzero vector orthogonal to the plane through the points $P , Q$ , and $R$ . Select the correct answer. $P ( 1,0,0 ) , Q ( 5,4,0 ) , R ( 0,4,1 )$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified