Question 1

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) B) and C)
G) A) and E)

Question 2

Find the area of the triangle with the given vertices.
 (−4,−2,2) ,(−9,−7,0) ,(−4,−3,5) ( - 4 , - 2,2 )  , ( - 9 , - 7,0 )  , ( - 4 , - 3,5 ) (−4,−2,2) ,(−9,−7,0) ,(−4,−3,5)

Accepted Answer

A)   0
B)   1
C)    7112\frac { 7 \sqrt { 11 } } { 2 }2711c-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)    7114\frac { 7 \sqrt { 11 } } { 4 }4711c-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)    7117 \sqrt { 11 }711c-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) C) and D)
G) A) and B) C

Question 3

Find the magnitude of the vector  v\mathbf { v }v .
 v=⟨−7,1,−8⟩\mathbf { v } = \langle - 7,1 , - 8 \ranglev=⟨−7,1,−8⟩

Accepted Answer

A)   - 14
B)   16
C)    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'/>​ 
D)    21142 \sqrt { 114 }2114c-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)    114\sqrt { 114 }114c-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) C) and E)
G) D) and E)

Question 4

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) C) and D)
G) None of the above

Question 5

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point  (−3,6,−2) ( - 3,6 , - 2 ) (−3,6,−2)   and is parallel to the  yzy zyz -plane

Accepted Answer

A)    y=6y = 6y=6 
B)    z=−2z = - 2z=−2 
C)    x+y+z=1x + y + z = 1x+y+z=1 
D)    y+z=4y + z = 4y+z=4 
E)    x=−3x = - 3x=−3F) B) and E)
G) B) and D)

Question 6

Find a set of parametric equations for the line that passes through the given points. Show all your work. $$\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)$$

Accepted Answer

Answers may vary. On...

Question 7

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. One possible answer is shown below.
$$x = 9 - 12 t , y = 8 - 14 t , z = - 2 + 6 t$$

Question 8

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line.
 x=−1+5tx = - 1 + 5 tx=−1+5t 
 (6,−8,9) ,y=−3+t( 6 , - 8,9 )  , \quad y = - 3 + t(6,−8,9) ,y=−3+t 
 z=−2−6tz = - 2 - 6 tz=−2−6t

Accepted Answer

A)    5x+y−6z=05 x + y - 6 z = 05x+y−6z=0 
B)    5x+y−6z−32=05 x + y - 6 z - 32 = 05x+y−6z−32=0 
C)    5x+y−6z+32=05 x + y - 6 z + 32 = 05x+y−6z+32=0 
D)    6x−8y+9z−32=06 x - 8 y + 9 z - 32 = 06x−8y+9z−32=0 
E)    6x−8y+9z+32=06 x - 8 y + 9 z + 32 = 06x−8y+9z+32=0F) B) and D)
G) A) and B)

Question 9

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 C)

Question 10

Find the angle between the two planes in degrees. Round to a tenth of a degree.
 2x−y−4z=4−3x−2y+4z=4\begin{array} { l } 2 x - y - 4 z = 4 \- 3 x - 2 y + 4 z = 4\end{array}2x−y−4z=4−3x−2y+4z=4​

Accepted Answer

A)    11.5∘11.5 ^ { \circ }11.5∘ 
B)    54.1∘54.1 ^ { \circ }54.1∘ 
C)    144.1∘144.1 ^ { \circ }144.1∘ 
D)    39.0∘39.0 ^ { \circ }39.0∘ 
E)    2.5∘2.5 ^ { \circ }2.5∘F) C) and D)
G) A) and E) C

Question 11

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. 
$x=-8+7 t$
$(-5,-3,3) \text {, parallel to } y=7+3 t$
$z=-9-5 t$

Accepted Answer

Answers may vary. On...

Question 12

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) C) and E)
G) B) and E)

Question 13

Find the volume of the parallelpiped with the given vertices.  A(8,−5,−4) ,B(2,2,5) ,C(9,4,−12) ,D(3,11,−3) E(8,−6,−10) ,F(2,1,−1) ,G(9,3,−18) ,H(3,10,−9) \begin{array} { l } A ( 8 , - 5 , - 4 )  , B ( 2,2,5 )  , C ( 9,4 , - 12 )  , D ( 3,11 , - 3 )  \E ( 8 , - 6 , - 10 )  , F ( 2,1 , - 1 )  , G ( 9,3 , - 18 )  , H ( 3,10 , - 9 ) \end{array}A(8,−5,−4) ,B(2,2,5) ,C(9,4,−12) ,D(3,11,−3) E(8,−6,−10) ,F(2,1,−1) ,G(9,3,−18) ,H(3,10,−9) ​  )

Accepted Answer

A)   369
B)   405
C)   321
D)   455
E)   430 F) A) and D)
G) A) and C)

Question 14

Find the center and radius of the sphere.
 x2+y2+z2−16x+6y+6z+18=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0x2+y2+z2−16x+6y+6z+18=0

Accepted Answer

A)   center:  (−8,3,3) ( - 8,3,3 ) (−8,3,3)  ; radius: 8
B)   center:  (8,−3,−3) ( 8 , - 3 , - 3 ) (8,−3,−3)  ; radius: 8
C)   center:  (8,3,3) ( 8,3,3 ) (8,3,3)  ; radius: 8
D)   center:  (−8,3,3) ( - 8,3,3 ) (−8,3,3)  ; radius: 64
E)   center:  (8,−3,−3) ( 8 , - 3 , - 3 ) (8,−3,−3)  ; radius: 64 F) C) and D)
G) B) and E)

Question 15

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

Accepted Answer

A)    (4,−3,9) ( 4 , - 3,9 ) (4,−3,9)  
B)    (4,9,−3) ( 4,9 , - 3 ) (4,9,−3)  
C)    (4,−9,−3) ( 4 , - 9 , - 3 ) (4,−9,−3)  
D)    (4,−3,−9) ( 4 , - 3 , - 9 ) (4,−3,−9)  
E)    (4,13,10) ( 4,13,10 ) (4,13,10) F) A) and B)
G) None of the above

Question 16

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) A) and D)
G) None of the above

Question 17

Find the triple scalar product  u⋅(v×w) \mathbf { u } \cdot ( \mathbf { v } 	imes \mathbf { w } ) u⋅(v×w)   for the vectors
 u=⟨−3,−5,−8⟩,v=⟨−1,−6,3⟩,w=⟨−8,−7,6⟩\mathbf { u } = \langle - 3 , - 5 , - 8 angle , \mathbf { v } = \langle - 1 , - 6,3 angle , \mathbf { w } = \langle - 8 , - 7,6 angleu=⟨−3,−5,−8⟩,v=⟨−1,−6,3⟩,w=⟨−8,−7,6⟩

Accepted Answer

A)    −378- 378−378 
B)   463
C)    −463- 463−463 
D)   283
E)   0 F) None of the above
G) A) and D)

Question 18

Find the magnitude of the vector  v\mathbf { v }v .
 v=⟨0,−4,−4⟩\mathbf { v } = \langle 0 , - 4 , - 4 \ranglev=⟨0,−4,−4⟩

Accepted Answer

A)    −8- 8−8 
B)   8
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)    828 \sqrt { 2 }82c-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)    424 \sqrt { 2 }42c-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 E)
G) B) and E)

Question 19

Write the component form of the vector described below.
Initial point:  (−7,2,−1) ( - 7,2 , - 1 ) (−7,2,−1)  
Terminal point:  (−5,−9,−6) ( - 5 , - 9 , - 6 ) (−5,−9,−6)

Accepted Answer

A)    ⟨2,−11,−5⟩\langle 2 , - 11 , - 5 \rangle⟨2,−11,−5⟩ 
B)    ⟨−2,11,5⟩\langle - 2,11,5 \rangle⟨−2,11,5⟩ 
C)    ⟨−5,−9,−6⟩\langle - 5 , - 9 , - 6 \rangle⟨−5,−9,−6⟩ 
D)    ⟨2,−11,−7⟩\langle 2 , - 11 , - 7 \rangle⟨2,−11,−7⟩ 
E)    ⟨35,−18,6⟩\langle 35 , - 18,6 \rangle⟨35,−18,6⟩F) C) and D)
G) A) and C)

Question 20

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. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$ )

Accepted Answer

Let point

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Exam 10: Analytic Geometry in Three Dimensions

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
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Find the area of the triangle with the given vertices. $( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )$

Correct Answer
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C

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 7,1 , - 8 \rangle$

Correct Answer
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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
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Find the general form of the equation of the plane with the given characteristics. The plane passes through the point $( - 3,6 , - 2 )$ and is parallel to the $y z$ -plane

Correct Answer
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Find a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)$

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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
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Answers may vary. One possible answer is shown below. \[x = 9 - 12 t , y = 8 - 14 t , z = - 2 + 6 t\]

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = - 1 + 5 t$ $( 6 , - 8,9 ) , \quad y = - 3 + t$ $z = - 2 - 6 t$

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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
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Find the angle between the two planes in degrees. Round to a tenth of a degree. $\begin{array} { l } 2 x - y - 4 z = 4 \\- 3 x - 2 y + 4 z = 4\end{array}$

Correct Answer
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C

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x=-8+7 t$ $(-5,-3,3) \text {, parallel to } y=7+3 t$ $z=-9-5 t$

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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.

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Find the volume of the parallelpiped with the given vertices. $\begin{array} { l } A ( 8 , - 5 , - 4 ) , B ( 2,2,5 ) , C ( 9,4 , - 12 ) , D ( 3,11 , - 3 ) \\E ( 8 , - 6 , - 10 ) , F ( 2,1 , - 1 ) , G ( 9,3 , - 18 ) , H ( 3,10 , - 9 ) \end{array}$ )

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Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$

Correct Answer
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Find the coordinates of the point located four units in front of the $y z$ -plane, nine units to the right of the $x z$ -plane, and three units below the $x y$ -plane.

Correct Answer
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Correct Answer
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Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = \langle - 3 , - 5 , - 8 \rangle , \mathbf { v } = \langle - 1 , - 6,3 \rangle , \mathbf { w } = \langle - 8 , - 7,6 \rangle$

Correct Answer
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Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 4 , - 4 \rangle$

Correct Answer
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Write the component form of the vector described below. Initial point: $( - 7,2 , - 1 )$ Terminal point: $( - 5 , - 9 , - 6 )$

Correct Answer
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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. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$ )

Correct Answer
verified
Let point , point , and point .
\[\begin ...
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Exam 10: Analytic Geometry in Three Dimensions

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

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Find the area of the triangle with the given vertices. (−4,−2,2) ,(−9,−7,0) ,(−4,−3,5) ( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 ) (−4,−2,2) ,(−9,−7,0) ,(−4,−3,5)

Correct AnswerverifiedC

Find the magnitude of the vector v\mathbf { v }v . v=⟨−7,1,−8⟩\mathbf { v } = \langle - 7,1 , - 8 \ranglev=⟨−7,1,−8⟩

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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 point (−3,6,−2) ( - 3,6 , - 2 ) (−3,6,−2) and is parallel to the yzy zyz -plane

Correct AnswerverifiedShow Answer

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,92,72),(32,12,9)\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)(9,29​,27​),(23​,21​,9)

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

Correct AnswerverifiedAnswers may vary. One possible answer is shown below. \[x = 9 - 12 t , y = 8 - 14 t , z = - 2 + 6 t\]

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. x=−1+5tx = - 1 + 5 tx=−1+5t (6,−8,9) ,y=−3+t( 6 , - 8,9 ) , \quad y = - 3 + t(6,−8,9) ,y=−3+t z=−2−6tz = - 2 - 6 tz=−2−6t

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Correct AnswerverifiedShow Answer

Find the angle between the two planes in degrees. Round to a tenth of a degree. 2x−y−4z=4−3x−2y+4z=4\begin{array} { l } 2 x - y - 4 z = 4 \\- 3 x - 2 y + 4 z = 4\end{array}2x−y−4z=4−3x−2y+4z=4​

Correct AnswerverifiedC

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. x=−8+7tx=-8+7 tx=−8+7t (−5,−3,3), parallel to y=7+3t(-5,-3,3) \text {, parallel to } y=7+3 t(−5,−3,3), parallel to y=7+3t z=−9−5tz=-9-5 tz=−9−5t

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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

Correct AnswerverifiedShow Answer

Find the center and radius of the sphere. x2+y2+z2−16x+6y+6z+18=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0x2+y2+z2−16x+6y+6z+18=0

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the magnitude of the vector v\mathbf { v }v . v=⟨0,−4,−4⟩\mathbf { v } = \langle 0 , - 4 , - 4 \ranglev=⟨0,−4,−4⟩

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Write the component form of the vector described below. Initial point: (−7,2,−1) ( - 7,2 , - 1 ) (−7,2,−1) Terminal point: (−5,−9,−6) ( - 5 , - 9 , - 6 ) (−5,−9,−6)

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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. (−9,6,−1),(−8,3,−2),(3,6,0)( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )(−9,6,−1),(−8,3,−2),(3,6,0) )

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

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 area of the triangle with the given vertices. $( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )$

Correct Answer
verified
C

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 7,1 , - 8 \rangle$

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 point $( - 3,6 , - 2 )$ and is parallel to the $y z$ -plane

Correct Answer
verified

Find a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)$

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

Correct Answer
verified
Answers may vary. One possible answer is shown below. \[x = 9 - 12 t , y = 8 - 14 t , z = - 2 + 6 t\]

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = - 1 + 5 t$ $( 6 , - 8,9 ) , \quad y = - 3 + t$ $z = - 2 - 6 t$

Correct Answer
verified

Correct Answer
verified

Find the angle between the two planes in degrees. Round to a tenth of a degree. $\begin{array} { l } 2 x - y - 4 z = 4 \\- 3 x - 2 y + 4 z = 4\end{array}$

Correct Answer
verified
C

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x=-8+7 t$ $(-5,-3,3) \text {, parallel to } y=7+3 t$ $z=-9-5 t$

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

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

Correct Answer
verified

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 4 , - 4 \rangle$

Correct Answer
verified

Write the component form of the vector described below. Initial point: $( - 7,2 , - 1 )$ Terminal point: $( - 5 , - 9 , - 6 )$

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. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$ )

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