Exam 10: Analytic Geometry in Three Dimensions

Find a unit vector orthogonal to $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } =$ leadcoeff(a)i $\operatorname { coeff } ( b ) \mathbf { j } \operatorname { coeff } ( \mathrm { c } ) \mathbf { k } , \mathbf { v } =$ leadcoeff(d) $\mathbf { c o e f f } ( \mathrm { f } ) \mathbf { j } \operatorname { coeff } ( \mathrm { g } ) \mathbf { k }$

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

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

Find the area of the triangle with the given vertices. $( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )$

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = - \mathbf { i } - 5 \mathbf { j } + 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } - 3 \mathbf { j } - 8 \mathbf { k } , \mathbf { w } = - \mathbf { i } - 6 \mathbf { j } + \mathbf { k }$

Find the volume of the parallelpiped with the given vertices. $A ( 0 , - 5,3 ) , B ( 3 , - 1 , - 4 ) , C ( 7 , - 3 , - 1 ) , D ( 10,1 , - 8 ) \text {, }$ $E ( - 3 , - 13,6 ) , F ( 0 , - 9 , - 1 ) , G ( 4 , - 11,2 ) , H ( 7 , - 7 , - 5 )$

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle - 3,3,9 \rangle$ and $\mathbf { v } = \langle - 7,5,7 \rangle$ . $\mathbf { z } = - 3 \mathbf { u } - 5 \mathbf { v }$

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

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.

Use the scalar triple product to find the volume of the parallelepiped having adjacent edges $\langle 1,3,5 \rangle , \langle 2,3,3 \rangle$ , and $\langle 3,3,4 \rangle$ .

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 magnitude of the vector $\mathbf { v }$ described below. Initial point: $( 1 , - 8 , - 6 )$ Terminal point: $( - 9 , - 1 , - 6 )$

Find the coordinates of the point located four units in front of the $y z$ -plane, nine units to the right of the $x z$ -plane, and three units below the $x y$ -plane.

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = \langle - 3 , - 2,4 \rangle , \mathbf { v } = \langle - 4 , - 5 , - 7 \rangle$

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

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = - 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } , \mathbf { v } = - 6 \mathbf { i } - 7 \mathbf { j } - 3 \mathbf { k }$

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point $( - 3,6 , - 2 )$ and is parallel to the $y z$ -plane.

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

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

The weight of a crate is 200 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. point $A = ( 0,0 , - 180 )$ , point $B = ( 110,0,0 )$ , point $C = ( - 40,50,0 )$ , point $D = ( 0 , - 130,0 )$