Exam 12: Vectors and the Geometry of Space

Calculate the angle between a and b (correct to the nearest degree). $\mathbf { a } = 9 \mathbf { i } - 8 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 8 z = 40$

The tension T at each end of the chain has magnitude $27$ N and makes an angle $\angle a = 27 ^ { \circ }$ with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , \quad z = 7 t$

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. $P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$

Find the equation of the line through $( 6 , - 6,6 )$ and perpendicular to the plane $- x + 2 y + 5 z = 12$

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is $\text { three }$ times the distance from P to the yz-plane.

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes $x = 3$ , $y = 6$ , and $z = 4$ .

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

Plot the given points in a three-dimensional coordinate system. (1, 2, 3)

Find that the midpoint of the line segment from $P 1 ( - 12,12,9 )$ to $P 2 ( 2,8,3 )$ .

Draw a rectangular box with the origin and $( 7,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

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

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)

Which of the given lines is parallel to the line $x = 8 + t , y = t , z = - 7 - 10 t ?$

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18$ , $| \mathbf { v } | = 3$ , $\angle \theta = 80 ^ { \circ }$ .

a. Find an equation of the sphere that passes through the point $( 6 , - 6,3 )$ and has center $( - 3,6,3 )$ . b. Find the curve in which this sphere intersects the xy-plane.

A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is $12$ cm long. Find the magnitude of the torque about P correct to two decimal places. Let $a = 66$ N, $b = 70 ^ { \circ }$ , $c = 10 ^ { \circ }$ .

Find an equation for the surface obtained by rotating the line $x = 7 z$ about the x-axis.

Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let $a = 3$ ft, $b = 3$ ft, $c = 30 ^ { \circ }$ , $d = 41$ lb.