Exam 10: Analytic Geometry in Three Dimensions

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle - 4,9,8 \rangle , \mathbf { v } = \langle 5,4,4 \rangle$ , and $\mathbf { w } = \langle - 3,23,4 \rangle$ . $3 \mathbf { u } - \mathbf { v } + 2 \mathbf { z } = \mathbf { w }$

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.]

Find a set of parametric equations for the line through the point and parallel to the specified vector. Show all your work. $( - 1 , - 5,9 )$ , parallel to $\langle - 7,6 , - 3 \rangle$

Determine the values of $c$ such that $\| c \mathbf { u } \| = 6$ , where $\mathbf { u } = - 3 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k }$ .

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangle$

Find the area of the parallelogram formed by the points $A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )$ , $C ( 3 , - 2,11 )$ , and $D ( 8 , - 1,13 )$ .

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 5 , - 3 \rangle$

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 a unit vector orthogonal to $3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { j } + 5 \mathbf { k }$ .

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

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.] $( 5,2 , - 1 ) , ( 3,3,2 ) , ( 4,4,1 )$

Find the magnitude of the vector described below. Initial point: $( 0 , - 6 , - 4 )$ Terminal point: $( - 1,5,5 )$

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. $\mathbf { u } = - 6 \mathbf { i } + 9 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = - 3 \mathbf { i } - 7 \mathbf { j } - 4 \mathbf { k }$

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

For the points $A ( 1 , - 2 , - 1 ) , B ( 2 , - 7,4 ) , C ( 1 , - 6 , - 5 ) , D ( 2 , - 11,0 )$ : a. Verify that the points are vertices of a parallelogram. Show all work. b. Find the area of the parallelogram. Show all work. c. Determine whether the parallelogram is a rectangle.

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = \langle 7 , - 5,1 \rangle , \mathbf { v } = \langle - 3,4,8 \rangle$

Find the volume of the parallelpiped with the given vertices. $A ( - 3 , - 8 , - 6 ) , B ( - 7,0,2 ) , C ( - 12 , - 3,3 ) , D ( - 16,5,11 )$ , $E ( - 9 , - 7,2 ) , F ( - 13,1,10 ) , G ( - 18 , - 2,11 ) , H ( - 22,6,19 )$

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

D Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = - 7 \mathbf { i } - 3 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }$