Exam 10: Analytic Geometry in Three Dimensions

Find a set of parametric equations for the line through the point and parallel to the specified line. Show all your work. $x = 9 - 8 t$ $( - 3 , - 7 , - 6 )$ , parallel to y=3-4t z=-2+6t

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.

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 angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+5z=3 -x+3y+3z=3

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

Find the distance between the point and the plane. (-2,-1,-4) -3x-2y-z=1

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle$ , and $\mathbf { w } = \langle - 13,34 , - 10 \rangle$ . $- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4 , - 1,4 \rangle , \mathbf { v } = \langle 2 , - 3,0 \rangle$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = \langle 1 , - 6 , - 4 \rangle , \mathbf { v } = \langle 6,4,9 \rangle$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = - 1 + 5 t$ $( 6 , - 8,9 ) , \quad y = - 3 + t$ $z = - 2 - 6 t$

The weight of a crate is 300 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 $A = ( 0,0 , - 130 )$ , point $B = ( 90,0,0 )$ , point $C = ( - 40,40,0 )$ , point $D = ( 0 , - 1$

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

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

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 the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 2 , - 3,6 ) , ( - 3 , - 8,5 ) , ( 2 , - 8,0 )$

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

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = \langle - 3 , - 5 , - 8 \rangle , \mathbf { v } = \langle - 1 , - 6,3 \rangle , \mathbf { w } = \langle - 8 , - 7,6 \rangle$