Question 1

Find a set of symmetric equations of the line that passes through the points  (5,0,5) ( 5,0,5 ) (5,0,5)   and  (7,8,−6) ( 7,8 , - 6 ) (7,8,−6)  .

Accepted Answer

A)    x7=y8=z−6\frac { x } { 7 } = \frac { y } { 8 } = \frac { z } { - 6 }7x​=8y​=−6z​ 
B)    x−52=y−88=z−6−11\frac { x - 5 } { 2 } = \frac { y - 8 } { 8 } = \frac { z - 6 } { - 11 }2x−5​=8y−8​=−11z−6​ 
C)    x−5−2=y8=z−511\frac { x - 5 } { - 2 } = \frac { y } { 8 } = \frac { z - 5 } { 11 }−2x−5​=8y​=11z−5​ 
D)    x+25=y−8=z−115\frac { x + 2 } { 5 } = y - 8 = \frac { z - 11 } { 5 }5x+2​=y−8=5z−11​ 
E)    x−52=y8=z−5−11\frac { x - 5 } { 2 } = \frac { y } { 8 } = \frac { z - 5 } { - 11 }2x−5​=8y​=−11z−5​F) None of the above
G) A) and D)

Question 2

Find the distance between the points.
 (1,9,3) ,(5,−6,8) ( 1,9,3 )  , ( 5 , - 6,8 ) (1,9,3) ,(5,−6,8)

Accepted Answer

A)   20
B)   24
C)    262 \sqrt { 6 }26c-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)    22662 \sqrt { 266 }2266c-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'/>​ 
E)    266\sqrt { 266 }266c-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'/>​F) A) and B)
G) A) and E)

Question 3

Find the area of the parallelogram that has the vectors as adjacent sides.
 u=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangleu=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩

Accepted Answer

A)   12
B)    41\sqrt { 41 }41c-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)    222 \sqrt { 2 }22c-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)   15
E)    5415 \sqrt { 41 }541c-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'/>​F) All of the above
G) A) and B)

Question 4

Find the coordinates of the point located three units in front of the  yzy zyz -plane, eight units to the right of the  xzx zxz -plane, and five units above the  xyx yxy -plane.

Accepted Answer

A)    (3,5,8) ( 3,5,8 ) (3,5,8)  
B)    (3,8,5) ( 3,8,5 ) (3,8,5)  
C)    (3,−8,5) ( 3 , - 8,5 ) (3,−8,5)  
D)    (3,5,−8) ( 3,5 , - 8 ) (3,5,−8)  
E)    (3,11,16) ( 3,11,16 ) (3,11,16) F) A) and B)
G) A) and E)

Question 5

Determine whether  u\mathbf { u }u  and  v\mathbf { v }v  are parallel, orthogonal, or neither.
 u=⟨5,6,−8⟩,v=⟨10,12,−16⟩\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangleu=⟨5,6,−8⟩,v=⟨10,12,−16⟩

Accepted Answer

A)   parallel
B)   orthogonal
C)   neither D) A) and B)
E) A) and C)

Question 6

Find the torque on the crankshaft  V\mathbf { V }V  using the data shown in the figure. Round to the nearest tenth of a foot-pound.

∥V∥=1.6ft∥F∥=20lbθ=60∘\begin{array} { l } \| \mathbf { V } \| = 1.6 \mathrm { ft } \\| \mathbf { F } \| = 20 \mathrm { lb } \	heta = 60 ^ { \circ }\end{array}∥V∥=1.6ft∥F∥=20lbθ=60∘​

Accepted Answer

A)    27.7ft−lb27.7 \mathrm { ft } - \mathrm { lb }27.7ft−lb 
B)    16.0ft−lb16.0 \mathrm { ft } - \mathrm { lb }16.0ft−lb 
C)    55.4ft−lb55.4 \mathrm { ft } - \mathrm { lb }55.4ft−lb 
D)    32.0ft−lb32.0 \mathrm { ft } - \mathrm { lb }32.0ft−lb 
E)   0 F) C) and D)
G) D) and E)

Question 7

Find the vector  z\mathbf { z }z , given  u=⟨5,−8,2⟩,v=⟨−6,−2,9⟩\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangleu=⟨5,−8,2⟩,v=⟨−6,−2,9⟩ , and  w=⟨−13,34,−10⟩\mathbf { w } = \langle - 13,34 , - 10 \ranglew=⟨−13,34,−10⟩ .  −3u−2v−2z=w- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }−3u−2v−2z=w

Accepted Answer

A)    ⟨−10,6,14⟩\langle - 10,6,14 \rangle⟨−10,6,14⟩ 
B)    ⟨−3,5,−7⟩\langle - 3,5 , - 7 \rangle⟨−3,5,−7⟩ 
C)    ⟨−5,3,−7⟩\langle - 5,3 , - 7 \rangle⟨−5,3,−7⟩ 
D)    ⟨5,−3,−7⟩\langle 5 , - 3 , - 7 \rangle⟨5,−3,−7⟩ 
E)    ⟨6,−3,−6⟩\langle 6 , - 3 , - 6 \rangle⟨6,−3,−6⟩F) B) and C)
G) A) and E)

Question 8

Determine the values of  ccc  such that  ∥cu∥=2\| c \mathbf { u } \| = 2∥cu∥=2 , where  u=3i−4j+2k\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }u=3i−4j+2k .

Accepted Answer

A)    c=±2c = \pm 2c=±2 
B)    c=±22929\quad c = \pm \frac { 2 \sqrt { 29 } } { 29 }c=±29229c-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)    c=±292c = \pm \frac { \sqrt { 29 } } { 2 }c=±229c-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)    c=±12c = \pm \frac { 1 } { 2 }c=±21​ 
E)    c=±2929c = \pm \frac { \sqrt { 29 } } { 29 }c=±2929c-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'/>​​F) A) and C)
G) A) and D)

Question 9

Find the angle between the two planes in degrees. Round to a tenth of a degree.
 x+3y+6z=4x−6y−z=4\begin{array} { l } x + 3 y + 6 z = 4 \x - 6 y - z = 4\end{array}x+3y+6z=4x−6y−z=4​

Accepted Answer

A)    9.4∘9.4 ^ { \circ }9.4∘ 
B)    33.4∘33.4 ^ { \circ }33.4∘ 
C)    123.4∘123.4 ^ { \circ }123.4∘ 
D)    28.8∘28.8 ^ { \circ }28.8∘ 
E)    2.2∘2.2 ^ { \circ }2.2∘F) D) and E)
G) A) and E)

Question 10

Find the coordinates of the point located five units in front of the  yzy zyz -plane, seven units to the right of the  xzx zxz -plane, and nine units below the  xyx yxy -plane.

Accepted Answer

A)    (5,−9,7) ( 5 , - 9,7 ) (5,−9,7)  
B)    (5,7,−9) ( 5,7 , - 9 ) (5,7,−9)  
C)    (5,−7,−9) ( 5 , - 7 , - 9 ) (5,−7,−9)  
D)    (5,−9,−7) ( 5 , - 9 , - 7 ) (5,−9,−7)  
E)    (5,12,3) ( 5,12,3 ) (5,12,3) F) B) and D)
G) A) and D)

Question 11

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work.
$$( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$$

Accepted Answer

Let point

, point

, and point

.
\[\begin ...

Question 12

Find the distance between the point and the plane.
 (−2,−1,−4) −3x−2y−z=1\begin{array} { l } ( - 2 , - 1 , - 4 )  \- 3 x - 2 y - z = 1\end{array}(−2,−1,−4) −3x−2y−z=1​

Accepted Answer

A)   11
B)    1114\frac { 11 } { 14 }1411​ 
C)    1314\frac { 13 } { \sqrt { 14 } }14c-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'/>​13​ 
D)   0
E)    1114\frac { 11 } { \sqrt { 14 } }14c-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'/>​11​F) C) and D)
G) C) and E)

Question 13

Find a set of parametric equations for the line that passes through the given points. Show all your work.
$$( 9,8 , - 2 ) , ( - 3 , - 6,4 )$$

Accepted Answer

Answers may vary. On...

Question 14

Find the dot product of  u\mathbf { u }u  and  v\mathbf { v }v .
 u=9i−5j−9k,v=−2i+6j+5k\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }u=9i−5j−9k,v=−2i+6j+5k

Accepted Answer

A)    −93- 93−93 
B)   4
C)    −18i−30j−45k- 18 \mathbf { i } - 30 \mathbf { j } - 45 \mathbf { k }−18i−30j−45k 
D)    −3- 3−3 
E)    7i+j−4k7 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }7i+j−4kF) D) and E)
G) B) and D)

Question 15

Find the area of the parallelogram that has the vectors as adjacent sides.
 u=⟨4,2,−5⟩,v=⟨−4,1,2⟩\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangleu=⟨4,2,−5⟩,v=⟨−4,1,2⟩

Accepted Answer

A)   14
B)    41\sqrt { 41 }41c-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)    262 \sqrt { 6 }26c-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)   11
E)    3413 \sqrt { 41 }341c-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'/>​F) B) and E)
G) A) and E)

Question 16

Find the general form of the equation of the plane with the given characteristics.
The plane passes through the points  (−4,4,3) ( - 4,4,3 ) (−4,4,3)   and  (2,7,−7) ( 2,7 , - 7 ) (2,7,−7)   and is perpendicular to the plane  2x+3y+3z=12 x + 3 y + 3 z = 12x+3y+3z=1 .

Accepted Answer

A)    39x−38y+12z+272=039 x - 38 y + 12 z + 272 = 039x−38y+12z+272=0 
B)    2x+3y+3z+13=02 x + 3 y + 3 z + 13 = 02x+3y+3z+13=0 
C)    39x+38y+12z+32=039 x + 38 y + 12 z + 32 = 039x+38y+12z+32=0 
D)    2x+3y+3z=02 x + 3 y + 3 z = 02x+3y+3z=0 
E)    38x−39y+14z+266=038 x - 39 y + 14 z + 266 = 038x−39y+14z+266=0F) B) and E)
G) B) and D)

Question 17

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work.
$$( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )$$

Accepted Answer

Let point

, point

, and point

.
\[\begin ...

Question 18

Find the general form of the equation of the plane passing through the three points. [Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.]
 (−1,−4,−2) ,(−6,5,4) ,(1,−2,−6) ( - 1 , - 4 , - 2 )  , ( - 6,5,4 )  , ( 1 , - 2 , - 6 ) (−1,−4,−2) ,(−6,5,4) ,(1,−2,−6)

Accepted Answer

A)    12x+2y+7z+34=012 x + 2 y + 7 z + 34 = 012x+2y+7z+34=0 
B)    12x+2y+7z=012 x + 2 y + 7 z = 012x+2y+7z=0 
C)    x+4y+2z=0x + 4 y + 2 z = 0x+4y+2z=0 
D)    12x+2y+7z−34=012 x + 2 y + 7 z - 34 = 012x+2y+7z−34=0 
E)    x+4y+2z−136=0x + 4 y + 2 z - 136 = 0x+4y+2z−136=0F) B) and D)
G) C) and D)

Question 19

Find the area of the parallelogram that has the vectors as adjacent sides.
 u=2i+j+k,v=5i−5j+5k\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }u=2i+j+k,v=5i−5j+5k

Accepted Answer

A)   5
B)    14\sqrt { 14 }14c-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)    10\sqrt { 10 }10c-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)   4
E)    5145 \sqrt { 14 }514c-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'/>​F) A) and B)
G) C) and D)

Question 20

The weight of a crate is 300 newtons. Find the tension in each of the supporting cables shown in the figure. The coordinates of the points  A,B,CA , B , CA,B,C , and  DDD  are given below the figure. Round to the nearest newton.

[Figure not necessarily to scale.]
point  A=(0,0,−130) A = ( 0,0 , - 130 ) A=(0,0,−130)  , point  B=(90,0,0) B = ( 90,0,0 ) B=(90,0,0)  , point  C=(−40,40,0) C = ( - 40,40,0 ) C=(−40,40,0)  , point  D=(0,−180,0) D = ( 0 , - 180,0 ) D=(0,−180,0)

Accepted Answer

A)   cable  AB=196A B = 196AB=196 ; cable  AC=97A C = 97AC=97 ; cable  AD=68A D = 68AD=68 
B)   cable  AB=97A B = 97AB=97 ; cable  AC=68A C = 68AC=68 ; cable  AD=196A D = 196AD=196 
C)   cable  AB=97A B = 97AB=97 ; cable  AC=196A C = 196AC=196 ; cable  AD=68A D = 68AD=68 
D)   cable  AB=68A B = 68AB=68 ; cable  AC=196A C = 196AC=196 ; cable  AD=97A D = 97AD=97 
E)   cable  AB=196A B = 196AB=196 ; cable  AC=68A C = 68AC=68 ; cable  AD=97A D = 97AD=97F) C) and D)
G) A) and B)

Exam 10: Analytic Geometry in Three Dimensions

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$

Correct Answer
verified
Let point , point , and point .
\[\begin ...
View Answer

Find the angle between the two planes in degrees. Round to a tenth of a degree. $\begin{array} { l } x + 3 y + 6 z = 4 \\x - 6 y - z = 4\end{array}$

Correct Answer
verified

Correct Answer
verified

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangle$

Correct Answer
verified

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( - 4,4,3 )$ and $( 2,7 , - 7 )$ and is perpendicular to the plane $2 x + 3 y + 3 z = 1$ .

Correct Answer
verified

Determine the values of $c$ such that $\| c \mathbf { u } \| = 2$ , where $\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$ .

Correct Answer
verified

Find the general form of the equation of the plane passing through the three points. [Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] $( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )$

Correct Answer
verified

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }$

Correct Answer
verified

Find the coordinates of the point located five units in front of the $y z$ -plane, seven units to the right of the $x z$ -plane, and nine units below the $x y$ -plane.

Correct Answer
verified

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )$

Correct Answer
verified
Let point , point , and point .
\[\begin ...
View Answer

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangle$

Correct Answer
verified

Find the distance between the point and the plane. $\begin{array} { l } ( - 2 , - 1 , - 4 ) \\- 3 x - 2 y - z = 1\end{array}$

Correct Answer
verified

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

Correct Answer
verified

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle$ , and $\mathbf { w } = \langle - 13,34 , - 10 \rangle$ . $- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }$

Correct Answer
verified

Correct Answer
verified

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$

Correct Answer
verified

Find the distance between the points. $( 1,9,3 ) , ( 5 , - 6,8 )$

Correct Answer
verified

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

Correct Answer
verified

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

Correct Answer
verified

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 9,8 , - 2 ) , ( - 3 , - 6,4 )$

Correct Answer
verified
Answers may vary. On...
View Answer

Exam 10: Analytic Geometry in Three Dimensions

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (−6,−3,4),(−7,−1,3),(6,4,0)( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )(−6,−3,4),(−7,−1,3),(6,4,0)

Correct Answerverified Let point , point , and point . \[\begin ...View AnswerShow Answer

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4x−6y−z=4\begin{array} { l } x + 3 y + 6 z = 4 \\x - 6 y - z = 4\end{array}x+3y+6z=4x−6y−z=4​

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangleu=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩

Correct AnswerverifiedShow Answer

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (−4,4,3) ( - 4,4,3 ) (−4,4,3) and (2,7,−7) ( 2,7 , - 7 ) (2,7,−7) and is perpendicular to the plane 2x+3y+3z=12 x + 3 y + 3 z = 12x+3y+3z=1 .

Correct AnswerverifiedShow Answer

Determine the values of ccc such that ∥cu∥=2\| c \mathbf { u } \| = 2∥cu∥=2 , where u=3i−4j+2k\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }u=3i−4j+2k .

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the area of the parallelogram that has the vectors as adjacent sides. u=2i+j+k,v=5i−5j+5k\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }u=2i+j+k,v=5i−5j+5k

Correct AnswerverifiedShow Answer

Find the coordinates of the point located five units in front of the yzy zyz -plane, seven units to the right of the xzx zxz -plane, and nine units below the xyx yxy -plane.

Correct AnswerverifiedShow Answer

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (8,−4,4),(9,5,3),(−3,6,0)( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )(8,−4,4),(9,5,3),(−3,6,0)

Correct Answerverified Let point , point , and point . \[\begin ...View AnswerShow Answer

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨4,2,−5⟩,v=⟨−4,1,2⟩\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangleu=⟨4,2,−5⟩,v=⟨−4,1,2⟩

Correct AnswerverifiedShow Answer

Find the distance between the point and the plane. (−2,−1,−4) −3x−2y−z=1\begin{array} { l } ( - 2 , - 1 , - 4 ) \\- 3 x - 2 y - z = 1\end{array}(−2,−1,−4) −3x−2y−z=1​

Correct AnswerverifiedShow Answer

Find a set of symmetric equations of the line that passes through the points (5,0,5) ( 5,0,5 ) (5,0,5) and (7,8,−6) ( 7,8 , - 6 ) (7,8,−6) .

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Determine whether u\mathbf { u }u and v\mathbf { v }v are parallel, orthogonal, or neither. u=⟨5,6,−8⟩,v=⟨10,12,−16⟩\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangleu=⟨5,6,−8⟩,v=⟨10,12,−16⟩

Correct AnswerverifiedShow Answer

Find the distance between the points. (1,9,3) ,(5,−6,8) ( 1,9,3 ) , ( 5 , - 6,8 ) (1,9,3) ,(5,−6,8)

Correct AnswerverifiedShow Answer

Find the dot product of u\mathbf { u }u and v\mathbf { v }v . u=9i−5j−9k,v=−2i+6j+5k\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }u=9i−5j−9k,v=−2i+6j+5k

Correct AnswerverifiedShow Answer

Find the coordinates of the point located three units in front of the yzy zyz -plane, eight units to the right of the xzx zxz -plane, and five units above the xyx yxy -plane.

Correct AnswerverifiedShow Answer

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,8,−2),(−3,−6,4)( 9,8 , - 2 ) , ( - 3 , - 6,4 )(9,8,−2),(−3,−6,4)

Correct AnswerverifiedAnswers may vary. On...View AnswerShow Answer

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$

Correct Answer
verified
Let point , point , and point .
\[\begin ...
View Answer

Find the angle between the two planes in degrees. Round to a tenth of a degree. $\begin{array} { l } x + 3 y + 6 z = 4 \\x - 6 y - z = 4\end{array}$

Correct Answer
verified

Correct Answer
verified

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangle$

Correct Answer
verified

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( - 4,4,3 )$ and $( 2,7 , - 7 )$ and is perpendicular to the plane $2 x + 3 y + 3 z = 1$ .

Correct Answer
verified

Determine the values of $c$ such that $\| c \mathbf { u } \| = 2$ , where $\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$ .

Correct Answer
verified

Correct Answer
verified

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }$

Correct Answer
verified

Find the coordinates of the point located five units in front of the $y z$ -plane, seven units to the right of the $x z$ -plane, and nine units below the $x y$ -plane.

Correct Answer
verified

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )$

Correct Answer
verified
Let point , point , and point .
\[\begin ...
View Answer

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangle$

Correct Answer
verified

Find the distance between the point and the plane. $\begin{array} { l } ( - 2 , - 1 , - 4 ) \\- 3 x - 2 y - z = 1\end{array}$

Correct Answer
verified

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$

Correct Answer
verified

Find the distance between the points. $( 1,9,3 ) , ( 5 , - 6,8 )$

Correct Answer
verified

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

Correct Answer
verified

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

Correct Answer
verified

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 9,8 , - 2 ) , ( - 3 , - 6,4 )$

Correct Answer
verified
Answers may vary. On...
View Answer