Exam 10: Analytic Geometry in Three Dimensions

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 0 , - 8 , - 4 \rangle , \mathbf { v } = \langle - 5,0 , - 6 \rangle$ , and $\mathbf { w } = \langle - 17,25 , - 1 \rangle$ . $- 2 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { z } = \mathbf { w }$

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. [Figure not necessarily to scale.] point $A = ( 0,0 , - 180 )$ , point $B = ( 110,0,0 )$ , point $C = ( - 40,50,0 )$ , point $D = ( 0 , - 130,0 )$

Find the cross product of the unit vectors $\mathrm { k } \times \mathrm { i }$ .

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

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

The lights in an auditorium are 25-pound disks of radius 16 inches. Each disk is supported by three equally spaced 50-inch wires attached to the ceiling. Find the Tension in each wire. Round your answer to two decimals.

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( - 4,4,3 )$ and $( 2,7 , - 7 )$ and is perpendicular to the plane $2 x + 3 y + 3 z = 1$ .

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

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

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+2y+6z=0 -4x-y-3z=0

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \mathbf { v } = 7 \mathbf { i } + 8 \mathbf { j } + 3 \mathbf { k } , \mathbf { w } = 4 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }$

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

Find the area of the parallelogram formed by the points $A ( 2 , - 3,5 ) , B ( 8 , - 2,7 )$ , $C ( 3 , - 2,8 )$ , and $D ( 9 , - 1,10 )$ .

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 the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = \langle - 2,6,6 \rangle , \mathbf { v } = \langle - 3,7,7 \rangle$

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

The lights in an auditorium are 30-pound disks of radius 24 inches. Each disk is supported by three equally spaced 45 -inch wires attached to the ceiling. Find the tension in each wire. Round your answer to two decimals.

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

Find the midpoint of the line segment joining the points. $( - 5,8,9 ) , ( - 8,9,2 )$

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = \langle - 1 , - 2,1 \rangle , \mathbf { v } = \langle - 8 , - 1 , - 8 \rangle$