Exam 10: Analytic Geometry in Three Dimensions

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

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. $\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangle$

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

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , \mathbf { v } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$

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 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 a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 4 , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right) , \left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , - 2 \right)$

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

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 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. $( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 2,3,5 \rangle , \mathbf { v } = \langle - 3,3 , - 5 \rangle$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( - 4,2,8 ) , ( 5 , - 2 , - 7 )$

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

Find the distance between the points. $( 2 , - 3,3 ) , ( - 6,2,7 )$

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

D Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle - 1,6 , - 3 \rangle , \mathbf { v } = \langle - 3,18 , - 9 \rangle$

Find a set of symmetric equations of the line that passes through the points $( 3,0,3 )$ and $( 7,4 , - 2 )$ .

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

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 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. $( - 2 , - 3,6 ) , ( - 3 , - 8,5 ) , ( 2 , - 8,0 )$