Exam 10: Analytic Geometry in Three Dimensions

Find the angle, in degrees, between two adjacent sides of the pyramid shown below. Round to the nearest tenth of a degree. [Note: The base of the pyramid is not considered a side.] $P(8,0,0), Q(8,8,0), R(0,8,0), S(4,4,3)$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = \langle - 8,7 , - 4 \rangle , \mathbf { v } = \langle 8,8,8 \rangle$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 4 , - 7,2 ) , ( - 7 , - 1 , - 5 )$

Find a set of symmetric equations of the line that passes through the points $( 6,0,6 )$ and $( 2,3 , - 7 )$ .

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

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.] $( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )$

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

Find the acute interior angle of the parallelogram formed by $A ( 1 , - 1,2 )$ , $B ( 6,0,4 ) , C ( 7,1,9 )$ , and $D ( 2,0,7 )$ . Round your answer to two decimals.

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = 4 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { v } = 7 \mathbf { i } - 7 \mathbf { j } + 4 \mathbf { k }$

The lights in an auditorium are 25-pound disks of radius 16 inches. Each disk is supported by three equally spaced 60 -inch wires attached to the ceiling. Find the tension in each wire. Round your answer to two decimals.

Find a set of parametric equations for the line through the point and parallel to the specified line. Show all your work. $x = - 7 - 7 t$ $( - 8 , - 4,3 )$ , parallel to y=2-8t z=-4+9t

Find the distance between the points. $( 1,5,5 ) , ( - 2,6 , - 1 )$

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = 6 \mathbf { i } - 3 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - \mathbf { i } - \mathbf { j } - 6 \mathbf { k }$

Find the distance between the point and the plane. (-3,-5,-2) 4x+6y+6z=-18

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle - 8,1,5 \rangle$ and $\mathbf { v } = \langle - 6 , - 3,6 \rangle$ . $\mathbf { z } = - 3 \mathbf { u } - 4 \mathbf { v }$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 6 , - 5,2 \rangle , \mathbf { v } = \langle 30 , - 25,10 \rangle$

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

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 7 , - 4,0 \rangle$

Find symmetric equations for the line through the point and parallel to the specified vector. Show all your work. $( - 5,3,8 )$ , parallel to $\langle - 4,5,7 \rangle$

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x = - 8 - 9 t$ $( 5 , - 9 , - 1 )$ , parallel to y=5-5t z=-8+3t