Exam 12: Vectors and the Geometry of Space

Calculate the angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree). $\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$ -plane.

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

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

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

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

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

A bicycle pedal is pushed by a foot with a force of $a - \mathrm { N }$ as shown. The shaft of the pedal is $13 \mathrm {~cm}$ long. Find the magnitude of the torque about $P$ correct to two decimal places. Let $a = 64 , b = 70 ^ { \circ } , c = 10 ^ { \circ }$ .

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 1,0,3 )$ to the point $( 4,2,7 )$ along a straight line. The distance is measured in meters and the force in newtons.

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \{ 5,5,3 \} , \mathbf { b } = \langle - 5,4 , - 4 \}$

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 4,8 , - 5 ) , B ( - 2 , - 8 , - 3 )$ and $C ( 12 , - 6,8 )$ .

Find the point of intersection. $L _ { 1 } : \frac { x - 17 } { 3 } = \frac { y - 58 } { 8 } = \frac { z - 23 } { 2 } , L _ { 2 } : \frac { x - 49 } { 7 } = \frac { y - 26 } { 4 } = z - 15$

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

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 9 \mathbf { j }$ and $\mathbf { b } = - 9 \mathbf { j } + 8 \mathbf { k }$ I. $\mathbf { a } \cdot \mathbf { b }$ II. $\mathbf { a } \times \mathbf { b }$ III. The angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree)

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 10 y = 0$

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 1,0,3 )$ to the point $( 5,3,8 )$ along a straight line. The distance is measured in meters and the force in newtons. Select the correct answer.

Find a nonzero vector orthogonal to the plane through the points $P , Q$ , and $R$ . Select the correct answer. $P ( 1,0,0 ) , Q ( 5,4,0 ) , R ( 0,4,1 )$