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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. (6,3,4),(7,1,3),(6,4,0)( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )

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Let point blured image, point blured image, and point blured image.
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Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4x6yz=4\begin{array} { l } x + 3 y + 6 z = 4 \\x - 6 y - z = 4\end{array}


A) 9.49.4 ^ { \circ }
B) 33.433.4 ^ { \circ }
C) 123.4123.4 ^ { \circ }
D) 28.828.8 ^ { \circ }
E) 2.22.2 ^ { \circ }

F) None of the above
G) A) and E)

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Find the torque on the crankshaft V\mathbf { V } using the data shown in the figure. Round to the nearest tenth of a foot-pound.  Find the torque on the crankshaft  \mathbf { V }  using the data shown in the figure. Round to the nearest tenth of a foot-pound.     \begin{array} { l }  \| \mathbf { V } \| = 1.6 \mathrm { ft } \\ \| \mathbf { F } \| = 20 \mathrm { lb } \\ \theta = 60 ^ { \circ } \end{array}  A)   27.7 \mathrm { ft } - \mathrm { lb }  B)   16.0 \mathrm { ft } - \mathrm { lb }  C)   55.4 \mathrm { ft } - \mathrm { lb }  D)   32.0 \mathrm { ft } - \mathrm { lb }  E)  0 V=1.6ftF=20lbθ=60\begin{array} { l } \| \mathbf { V } \| = 1.6 \mathrm { ft } \\\| \mathbf { F } \| = 20 \mathrm { lb } \\\theta = 60 ^ { \circ }\end{array}


A) 27.7ftlb27.7 \mathrm { ft } - \mathrm { lb }
B) 16.0ftlb16.0 \mathrm { ft } - \mathrm { lb }
C) 55.4ftlb55.4 \mathrm { ft } - \mathrm { lb }
D) 32.0ftlb32.0 \mathrm { ft } - \mathrm { lb }
E) 0

F) B) and D)
G) All of the above

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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 \rangle


A) 12
B) 41\sqrt { 41 }
C) 222 \sqrt { 2 }
D) 15
E) 5415 \sqrt { 41 }

F) A) and D)
G) C) and D)

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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 ) and (2,7,7) ( 2,7 , - 7 ) and is perpendicular to the plane 2x+3y+3z=12 x + 3 y + 3 z = 1 .


A) 39x38y+12z+272=039 x - 38 y + 12 z + 272 = 0
B) 2x+3y+3z+13=02 x + 3 y + 3 z + 13 = 0
C) 39x+38y+12z+32=039 x + 38 y + 12 z + 32 = 0
D) 2x+3y+3z=02 x + 3 y + 3 z = 0
E) 38x39y+14z+266=038 x - 39 y + 14 z + 266 = 0

F) None of the above
G) A) and E)

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Determine the values of cc such that cu=2\| c \mathbf { u } \| = 2 , where u=3i4j+2k\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } .


A) c=±2c = \pm 2
B) c=±22929\quad c = \pm \frac { 2 \sqrt { 29 } } { 29 }
C) c=±292c = \pm \frac { \sqrt { 29 } } { 2 }
D) c=±12c = \pm \frac { 1 } { 2 }
E) c=±2929c = \pm \frac { \sqrt { 29 } } { 29 }

F) B) and D)
G) A) and D)

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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 )


A) 12x+2y+7z+34=012 x + 2 y + 7 z + 34 = 0
B) 12x+2y+7z=012 x + 2 y + 7 z = 0
C) x+4y+2z=0x + 4 y + 2 z = 0
D) 12x+2y+7z34=012 x + 2 y + 7 z - 34 = 0
E) x+4y+2z136=0x + 4 y + 2 z - 136 = 0

F) A) and C)
G) C) and D)

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Find the area of the parallelogram that has the vectors as adjacent sides. u=2i+j+k,v=5i5j+5k\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }


A) 5
B) 14\sqrt { 14 }
C) 10\sqrt { 10 }
D) 4
E) 5145 \sqrt { 14 }

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

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


A) (5,9,7) ( 5 , - 9,7 )
B) (5,7,9) ( 5,7 , - 9 )
C) (5,7,9) ( 5 , - 7 , - 9 )
D) (5,9,7) ( 5 , - 9 , - 7 )
E) (5,12,3) ( 5,12,3 )

F) A) and D)
G) B) and D)

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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. (8,4,4),(9,5,3),(3,6,0)( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )

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Let point blured image, point blured image, and point blured image.
\[\begin ...

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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 \rangle


A) 14
B) 41\sqrt { 41 }
C) 262 \sqrt { 6 }
D) 11
E) 3413 \sqrt { 41 }

F) C) and D)
G) A) and E)

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Find the distance between the point and the plane. (2,1,4) 3x2yz=1\begin{array} { l } ( - 2 , - 1 , - 4 ) \\- 3 x - 2 y - z = 1\end{array}


A) 11
B) 1114\frac { 11 } { 14 }
C) 1314\frac { 13 } { \sqrt { 14 } }
D) 0
E) 1114\frac { 11 } { \sqrt { 14 } }

F) A) and E)
G) A) and C)

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


A) x7=y8=z6\frac { x } { 7 } = \frac { y } { 8 } = \frac { z } { - 6 }
B) x52=y88=z611\frac { x - 5 } { 2 } = \frac { y - 8 } { 8 } = \frac { z - 6 } { - 11 }
C) x52=y8=z511\frac { x - 5 } { - 2 } = \frac { y } { 8 } = \frac { z - 5 } { 11 }
D) x+25=y8=z115\frac { x + 2 } { 5 } = y - 8 = \frac { z - 11 } { 5 }
E) x52=y8=z511\frac { x - 5 } { 2 } = \frac { y } { 8 } = \frac { z - 5 } { - 11 }

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

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Find the vector z\mathbf { z } , given u=5,8,2,v=6,2,9\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle , and w=13,34,10\mathbf { w } = \langle - 13,34 , - 10 \rangle . 3u2v2z=w- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }


A) 10,6,14\langle - 10,6,14 \rangle
B) 3,5,7\langle - 3,5 , - 7 \rangle
C) 5,3,7\langle - 5,3 , - 7 \rangle
D) 5,3,7\langle 5 , - 3 , - 7 \rangle
E) 6,3,6\langle 6 , - 3 , - 6 \rangle

F) A) and C)
G) A) and D)

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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 , C , and DD are given below the figure. Round to the nearest newton.  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 , C , and  D  are given below the figure. Round to the nearest newton.    [Figure not necessarily to scale.] point  A = ( 0,0 , - 130 )  , point  B = ( 90,0,0 )  , point  C = ( - 40,40,0 )  , point  D = ( 0 , - 180,0 )   A)  cable  A B = 196 ; cable  A C = 97 ; cable  A D = 68  B)  cable  A B = 97 ; cable  A C = 68 ; cable  A D = 196  C)  cable  A B = 97 ; cable  A C = 196 ; cable  A D = 68  D)  cable  A B = 68 ; cable  A C = 196 ; cable  A D = 97  E)  cable  A B = 196 ; cable  A C = 68 ; cable  A D = 97 [Figure not necessarily to scale.] point A=(0,0,130) A = ( 0,0 , - 130 ) , point B=(90,0,0) B = ( 90,0,0 ) , point C=(40,40,0) C = ( - 40,40,0 ) , point D=(0,180,0) D = ( 0 , - 180,0 )


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

F) A) and E)
G) B) and E)

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Determine whether u\mathbf { u } and v\mathbf { 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 \rangle


A) parallel
B) orthogonal
C) neither

D) All of the above
E) B) and C)

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


A) 20
B) 24
C) 262 \sqrt { 6 }
D) 22662 \sqrt { 266 }
E) 266\sqrt { 266 }

F) A) and B)
G) All of the above

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Find the dot product of u\mathbf { u } and v\mathbf { v } . u=9i5j9k,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 }


A) 93- 93
B) 4
C) 18i30j45k- 18 \mathbf { i } - 30 \mathbf { j } - 45 \mathbf { k }
D) 3- 3
E) 7i+j4k7 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }

F) A) and E)
G) A) and D)

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


A) (3,5,8) ( 3,5,8 )
B) (3,8,5) ( 3,8,5 )
C) (3,8,5) ( 3 , - 8,5 )
D) (3,5,8) ( 3,5 , - 8 )
E) (3,11,16) ( 3,11,16 )

F) A) and D)
G) All of the above

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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)( 9,8 , - 2 ) , ( - 3 , - 6,4 )

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