Question 1

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 5 \mathbf { j }$ and $\mathbf { b } = - 5 \mathbf { j } + 2 \mathbf { k }$.
I. $\mathbf { a } \cdot \mathbf { b }$
П. $\mathbf { a } \times \mathbf { b }$
II. 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 2

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

Accepted Answer

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

Question 3

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 4

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.

Accepted Answer

A)    4.794.794.79 
B)    6.796.796.79 
C)    4.294.294.29 
D)    7.797.797.79 
E)    5.795.795.79

Question 5

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 .

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 6

Find the center and the radius of the sphere that has the given equation.  x2+y2+z2−4x+8y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0x2+y2+z2−4x+8y=0

Accepted Answer

A)    (2,−4,0) ,25( 2 , - 4,0 )  , 2 \sqrt { 5 }(2,−4,0) ,25c-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'/>​ 
B)    (−2,4,0) ,25( - 2,4,0 )  , 2 \sqrt { 5 }(−2,4,0) ,25c-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)    (−2,4,0) ,20( - 2,4,0 )  , 20(−2,4,0) ,20 
D)    (2,−4,0) ,20( 2 , - 4,0 )  , 20(2,−4,0) ,20

Question 7

Sketch the plane in a three-dimensional space represented by the equation.  z=−1z = - 1z=−1

Accepted Answer

A)  
B)  
C)  
D)

Question 8

$\text { Write inequalities to describe the solid upper hemisphere of the sphere of radius } 3 \text { centered at the origin. }$

Accepted Answer

The answer of \(\text { Write inequalities to describe the...

Question 9

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

Accepted Answer

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

Question 10

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  (3,0,5) ( 3,0,5 ) (3,0,5)   to the point  (7,5,10) ( 7,5,10 ) (7,5,10)   along a straight line. The distance is measured in meters and the force in newtons.

Accepted Answer

A)    64 J64 \mathrm {~J}64 J 
B)    74 J74 \mathrm {~J}74 J 
C)    84 J84 \mathrm {~J}84 J 
D)    94 J94 \mathrm {~J}94 J 
E)    104 J104 \mathrm {~J}104 J

Question 11

Find an equation for the surface consisting of all points  PPP  for which the distance from  PPP  to the  xxx -axis is two times the distance from  PPP  to the  yzy zyz -plane.

Accepted Answer

A)    2x2=y2−2z22 x ^ { 2 } = y ^ { 2 } - 2 z ^ { 2 }2x2=y2−2z2 
B)    4x2=y2+z24 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }4x2=y2+z2 
C)    2x2=y2+2z2 x ^ { 2 } = y ^ { 2 } + 2 z2x2=y2+2z 
D)    x2−y2−z2=2x ^ { 2 } - y ^ { 2 } - z ^ { 2 } = 2x2−y2−z2=2 
E)    x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4

Question 12

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 13

Find the distance from  (−10,2,−8) ( - 10,2 , - 8 ) (−10,2,−8)   to the  xyx yxy -planes.

Accepted Answer

A)   8
B)   16
C)   10
D)   2
E)   4

Question 14

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

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 15

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$.

Accepted Answer

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

Question 16

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) A ( - 1,0,1 )  , B ( 1,1 , - 1 )  , C ( 1,1,1 ) A(−1,0,1) ,B(1,1,−1) ,C(1,1,1)

Accepted Answer

A)    3,2,53,2 , \sqrt { 5 }3,2,5c-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
B)    1,2,51,2 , \sqrt { 5 }1,2,5c-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)    1,1,21,1 , \sqrt { 2 }1,1,2c-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
D)    1,1,21,1 , \sqrt { 2 }1,1,2c-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 17

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)    1330\frac { 1 } { 3 } \sqrt { 30 }31​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 18

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 19

Find the cross product  a×b\mathbf { a } 	imes \mathbf { b }a×b .
 a=⟨7,5,1},b=⟨−5,2,−2}\mathbf { a } = \langle 7,5,1 \} , \mathbf { b } = \langle - 5,2 , - 2 \}a=⟨7,5,1},b=⟨−5,2,−2}

Accepted Answer

A)    (−8,−12,−19) ( - 8 , - 12 , - 19 ) (−8,−12,−19)  
B)    {−12,39,9}\{ - 12,39,9 \}{−12,39,9} 
C)    {−12,9,39) \{ - 12,9,39 ) {−12,9,39)  
D)    {39,−12,9}\{ 39 , - 12,9 \}{39,−12,9} 
E)    {−7,−23,25}\{ - 7 , - 23,25 \}{−7,−23,25}

Question 20

Find the center and radius of the sphere.  x2−6x−26+y2+24y+z2−18z=0x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0x2−6x−26+y2+24y+z2−18z=0

Accepted Answer

A)    C(−1,−12,−13) ,r=62C ( - 1 , - 12 , - 13 )  , r = 6 \sqrt { 2 }C(−1,−12,−13) ,r=62c-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'/>​ 
B)    C(−1,−12,−13) ,r=2C ( - 1 , - 12 , - 13 )  , r = \sqrt { 2 }C(−1,−12,−13) ,r=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'/>​ 
C)    C(1,−12,−13) ,r=6C ( 1 , - 12 , - 13 )  , r = 6C(1,−12,−13) ,r=6 
D)    C(−1,−12,−13) ,r=6C ( - 1 , - 12 , - 13 )  , r = 6C(−1,−12,−13) ,r=6 
E)   none of these

Exam 12: Vectors and the Geometry of Space

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

Correct Answer
verified

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

Correct Answer
verified

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

Correct Answer
verified

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

Correct Answer
verified

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

Correct Answer
verified

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$

Correct Answer
verified

$\text { Write inequalities to describe the solid upper hemisphere of the sphere of radius } 3 \text { centered at the origin. }$

Correct Answer
verified

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

Correct Answer
verified

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

Correct Answer
verified

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

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

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

Correct Answer
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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 an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

Correct Answer
verified

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.

Correct Answer
verified

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 $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Correct Answer
verified

Exam 12: Vectors and the Geometry of Space

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

Correct AnswerverifiedShow Answer

Sketch the plane in a three-dimensional space represented by the equation. z=−1z = - 1z=−1

Correct AnswerverifiedShow Answer

Find the distance from (−10,2,−8) ( - 10,2 , - 8 ) (−10,2,−8) to the xyx yxy -planes.

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Find the center and radius of the sphere. x2−6x−26+y2+24y+z2−18z=0x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0x2−6x−26+y2+24y+z2−18z=0

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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

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Find the center and the radius of the sphere that has the given equation. x2+y2+z2−4x+8y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0x2+y2+z2−4x+8y=0

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Find the length of the median of side ABA BAB of the triangle with vertices A(−8,6,−9),B(−6,−6,−5)A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )A(−8,6,−9),B(−6,−6,−5) and C(8,−6,2)C ( 8 , - 6,2 )C(8,−6,2) .

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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 .

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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

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

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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)

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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)

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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

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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) A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 ) A(−1,0,1) ,B(1,1,−1) ,C(1,1,1)

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Find an equation for the surface consisting of all points PPP for which the distance from PPP to the xxx -axis is two times the distance from PPP to the yzy zyz -plane.

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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.

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Correct AnswerverifiedShow Answer

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

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Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

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

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Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

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

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Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$

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Correct Answer
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Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

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

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

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

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

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

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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. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

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Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

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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.

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Correct Answer
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Correct Answer
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