Question 1

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)   1, 1,  2\sqrt { 2 }2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​  , both
B)   1, 2,  5\sqrt { 5 }5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​  , neither
C)   3, 2,  5\sqrt { 5 }5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​  , right
D)   1, 1,  2\sqrt { 2 }2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​  , isosceles

Question 2

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

Accepted Answer

A)    69.17969.17969.179  .
B)    69.17969.17969.179 
C)    53.17953.17953.179 .
D)    75.17975.17975.179 
E)    73.17973.17973.179

Question 3

Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let $a = 3$ ft, $b = 3$ ft, $c = 30 ^ { \circ }$ , $d = 41$ lb.

Accepted Answer

The answer of Find the magnitude of the torque about...

Question 4

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is   three 	ext { three } three   times the distance from P to the yz-plane.

Accepted Answer

A)    9x2=y2+z29 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }9x2=y2+z2 
B)    3x2=y2+3z3 x ^ { 2 } = y ^ { 2 } + 3 z3x2=y2+3z 
C)    x2−y2−z2=3x ^ { 2 } - y ^ { 2 } - z ^ { 2 } = 3x2−y2−z2=3 
D)    3x2=y2−3z23 x ^ { 2 } = y ^ { 2 } - 3 z ^ { 2 }3x2=y2−3z2 
E)    x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9

Question 5

Find the distance (correct to two decimal places)  between the given parallel planes.  6x+7y−2z=12 and 14x+16y−8z=406 x + 7 y - 2 z = 12 	ext { and } 14 x + 16 y - 8 z = 406x+7y−2z=12 and 14x+16y−8z=40

Accepted Answer

A)    10.9410.9410.94 
B)    1.941.941.94 
C)    3.943.943.94 
D)    5.945.945.94 
E)    7.947.947.94 B

Question 6

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

Accepted Answer

A)    x=−5+3t,y=10+t,z=−2−tx = - 5 + 3 t , y = 10 + t , z = - 2 - tx=−5+3t,y=10+t,z=−2−t 
B)    x+8=y−2=−9−zx + 8 = y - 2 = - 9 - zx+8=y−2=−9−z 
C)    x=−5t,y=10t,z=−1−tx = - 5 t , y = 10 t , z = - 1 - tx=−5t,y=10t,z=−1−t 
D)    x=−5+t,y=10+t,z=−1−tx = - 5 + t , y = 10 + t , z = - 1 - tx=−5+t,y=10+t,z=−1−t 
E)    r=⟨−6,−10,0}+t(2,2,−20) r = \langle - 6 , - 10,0 \} + t ( 2,2 , - 20 ) r=⟨−6,−10,0}+t(2,2,−20)

Question 7

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

Accepted Answer

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

Question 8

Write inequalities to describe the solid rectangular box in the first octant bounded by the
planes $x = 3$ , $y = 6$ , and $z = 4$ .

Accepted Answer

The answer of Write inequalities to describe the solid rectangular...

Question 9

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

Accepted Answer

A)    16\frac { 1 } { 6 }61​ 
B)    13\frac { 1 } { 3 }31​   30\sqrt { 30 }30c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
C)    5306\frac { 5 \sqrt { 30 } } { 6 }6530c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
D)    56\frac { 5 } { 6 }65​ 
E)    30\sqrt { 30 }30c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​

Question 10

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

Accepted Answer

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

Question 11

Calculate the angle between a and b (correct to the nearest degree) .  a=9i−8j,b=−7j+k\mathbf { a } = 9 \mathbf { i } - 8 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }a=9i−8j,b=−7j+k

Accepted Answer

A)    98∘98 ^ { \circ }98∘ 
B)    49∘49 ^ { \circ }49∘ 
C)    39∘39 ^ { \circ }39∘ 
D)    59∘59 ^ { \circ }59∘ 
E)   None of these  B

Question 12

Find the equation of the line through $( 6 , - 6,6 )$ and perpendicular to the plane $- x + 2 y + 5 z = 12$

Accepted Answer

The answer of Find the equation of the line through...

Question 13

The tension T at each end of the chain has magnitude $27$ N and makes an angle $\angle a = 27 ^ { \circ }$ with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.

Accepted Answer

$51.64$

Question 14

Plot the given points in a three-dimensional coordinate system. (1, 2, 3)

Accepted Answer

A)   
B)   
C)   
D)

Question 15

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

Accepted Answer

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

Question 16

Find that the midpoint of the line segment from  P1(−12,12,9) P 1 ( - 12,12,9 ) P1(−12,12,9)   to  P2(2,8,3) P 2 ( 2,8,3 ) P2(2,8,3)   .

Accepted Answer

A)    (−5,−10,−6) ( - 5 , - 10 , - 6 ) (−5,−10,−6)  
B)    (−5,10,6) ( - 5,10,6 ) (−5,10,6)  
C)    (−5,−10,6) ( - 5 , - 10,6 ) (−5,−10,6)  
D)    (−5,6,10) ( - 5,6,10 ) (−5,6,10)  
E)    (−5,10,−6) ( - 5,10 , - 6 ) (−5,10,−6)

Question 17

A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is $12$ cm long.
Find the magnitude of the torque about P correct to two decimal places.
Let $a = 66$ N, $b = 70 ^ { \circ }$ , $c = 10 ^ { \circ }$ .

Accepted Answer

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

Question 18

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

Accepted Answer

A)    (38,−10,−28) ( 38 , - 10 , - 28 ) (38,−10,−28)  
B)    (−43,−10,18) ( - 43 , - 10,18 ) (−43,−10,18)  
C)    (−38,−10,28) ( - 38 , - 10,28 ) (−38,−10,28)  
D)    (43,−10,−18) ( 43 , - 10 , - 18 ) (43,−10,−18)  
E)    (38,−10,28) ( 38 , - 10,28 ) (38,−10,28)

Question 19

Find an equation for the surface obtained by rotating the line $x = 7 z$ about the x-axis.

Accepted Answer

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

Question 20

Draw a rectangular box with the origin and $( 7,6,3 )$ 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...

Exam 12: Vectors and the Geometry of Space

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)

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is $\text { three }$ times the distance from P to the yz-plane.

Correct Answer
verified

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 8 z = 40$

Correct Answer
verified
B

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

Correct Answer
verified

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

Correct Answer
verified

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes $x = 3$ , $y = 6$ , and $z = 4$ .

Correct Answer
verified

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

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

Calculate the angle between a and b (correct to the nearest degree) . $\mathbf { a } = 9 \mathbf { i } - 8 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

Correct Answer
verified
B

Find the equation of the line through $( 6 , - 6,6 )$ and perpendicular to the plane $- x + 2 y + 5 z = 12$

Correct Answer
verified

Correct Answer
verified
\(51.64\)

Plot the given points in a three-dimensional coordinate system. (1, 2, 3)

Correct Answer
verified

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

Correct Answer
verified

Find that the midpoint of the line segment from $P 1 ( - 12,12,9 )$ to $P 2 ( 2,8,3 )$ .

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

Find an equation for the surface obtained by rotating the line $x = 7 z$ about the x-axis.

Correct Answer
verified

Draw a rectangular box with the origin and $( 7,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Correct Answer
verified

Exam 12: Vectors and the Geometry of Space

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)

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let a=3a = 3a=3 ft, b=3b = 3b=3 ft, c=30∘c = 30 ^ { \circ }c=30∘ , d=41d = 41d=41 lb.

Correct AnswerverifiedShow Answer

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is three \text { three } three times the distance from P to the yz-plane.

Correct AnswerverifiedShow Answer

Find the distance (correct to two decimal places) between the given parallel planes. 6x+7y−2z=12 and 14x+16y−8z=406 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 8 z = 406x+7y−2z=12 and 14x+16y−8z=40

Correct AnswerverifiedB

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

Correct AnswerverifiedShow Answer

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 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)

Correct AnswerverifiedShow Answer

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes x=3x = 3x=3 , y=6y = 6y=6 , and z=4z = 4z=4 .

Correct AnswerverifiedShow 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

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

Correct AnswerverifiedShow Answer

Calculate the angle between a and b (correct to the nearest degree) . a=9i−8j,b=−7j+k\mathbf { a } = 9 \mathbf { i } - 8 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }a=9i−8j,b=−7j+k

Correct AnswerverifiedB

Find the equation of the line through (6,−6,6)( 6 , - 6,6 )(6,−6,6) and perpendicular to the plane −x+2y+5z=12- x + 2 y + 5 z = 12−x+2y+5z=12

Correct AnswerverifiedShow Answer

The tension T at each end of the chain has magnitude 272727 N and makes an angle ∠a=27∘\angle a = 27 ^ { \circ }∠a=27∘ with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.

Correct Answerverified\(51.64\)

Plot the given points in a three-dimensional coordinate system. (1, 2, 3)

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Find that the midpoint of the line segment from P1(−12,12,9) P 1 ( - 12,12,9 ) P1(−12,12,9) to P2(2,8,3) P 2 ( 2,8,3 ) P2(2,8,3) .

Correct AnswerverifiedShow Answer

A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is 121212 cm long. Find the magnitude of the torque about P correct to two decimal places. Let a=66a = 66a=66 N, b=70∘b = 70 ^ { \circ }b=70∘ , c=10∘c = 10 ^ { \circ }c=10∘ .

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Find an equation for the surface obtained by rotating the line x=7zx = 7 zx=7z about the x-axis.

Correct AnswerverifiedShow Answer

Draw a rectangular box with the origin and (7,6,3)( 7,6,3 )(7,6,3) as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Correct AnswerverifiedShow Answer

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is $\text { three }$ times the distance from P to the yz-plane.

Correct Answer
verified

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 8 z = 40$

Correct Answer
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B

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

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

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Write inequalities to describe the solid rectangular box in the first octant bounded by the planes $x = 3$ , $y = 6$ , and $z = 4$ .

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

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

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Calculate the angle between a and b (correct to the nearest degree) . $\mathbf { a } = 9 \mathbf { i } - 8 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

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B

Find the equation of the line through $( 6 , - 6,6 )$ and perpendicular to the plane $- x + 2 y + 5 z = 12$

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\(51.64\)

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

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Find that the midpoint of the line segment from $P 1 ( - 12,12,9 )$ to $P 2 ( 2,8,3 )$ .

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

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Find an equation for the surface obtained by rotating the line $x = 7 z$ about the x-axis.

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Draw a rectangular box with the origin and $( 7,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

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