Exam 13: Vectors and the Geometry of Space

$\text { Calculate the direction of } \overrightarrow { P _ { 1 } P _ { 2 } } \text { and the midpoint of line segment } P _ { 1 } P _ { 2 } \text {. }$ - $\mathrm { P } _ { 1 } ( - 6 , - 7,4 )$ and $\mathrm { P } _ { 2 } ( - 5 , - 8,3 )$

$\text { Find the distance between points } P _ { 1 } \text { and } P _ { 2 } \text {. }$ - $\mathrm { P } _ { 1 } ( 7 , - 2,5 )$ and $\mathrm { P } _ { 2 } ( 10 , - 6,0 )$

$\text { Find the distance between points } P _ { 1 } \text { and } P _ { 2 } \text {. }$ - $P _ { 1 } ( 7,8 , - 7 )$ and $P _ { 2 } ( 9,5 , - 13 )$

Find the angle between u and v in radians. -u = -4j and v = 5i - 6k

Find the indicated vector. -Let $\mathbf { u } = \langle - 1 , - 8 \rangle , \mathbf { v } = \langle 4 , - 1 \rangle$ . Find $- 5 \mathbf { u } + 4 \mathbf { v }$ .

Find an equation for the sphere with the given center and radius. -Center (0, 0, 10), radius = 8

Find v · u. -v = 7i - 5j and u = -3i - 9j

Solve the problem. -Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2) and R(2, 2, 1).

Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin. -u = i - j, v = k

Find parametric equations for the line described below. -The line through the point P( , , 5) parallel to the vector -4i + 3j - 7k

Determine whether the following is always true or not always true. Given reasons for your answers. -c(u × v) = cu × cv (any number c)

$\text { Use the vectors } u , v , w \text {, and } z \text { head to tail as needed to sketch the indicated vector. }$ - $\mathbf { V } = \mathbf { W }$

Describe the given set of points with a single equation or with a pair of equations. -The set of points equidistant from the points (0, 0, -6) and (0, 0, 4)

Provide an appropriate response. -The unit vectors $\mathbf { u }$ and $\mathbf { v }$ are combined to produce two new vectors $\mathbf { a } = \mathbf { u } + \mathbf { v }$ and $\mathbf { b } = \mathbf { u } - \mathbf { v }$ . Show that $\mathbf { a }$ and $\mathbf { b }$ are orthogonal. Assume $\mathbf { u } \neq \mathbf { v }$ .

Find the indicated vector. -Let $\mathbf { u } = \langle - 1 , - 9 \rangle , \mathbf { v } = \langle 1 , - 3 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Calculate the requested distance. -The distance from the point S(3, 2, -6) to the line x = -6 + 2t, y = 9 + 2t, z = -8 + 2t

Find the vector projv u. -v = k, u = 9i + 2j + 6k

Give a geometric description of the set of points whose coordinates satisfy the given conditions. - $\mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 16 , \mathrm { x } = - 5$

$\text { Use the vectors } u , v , w \text {, and } z \text { head to tail as needed to sketch the indicated vector. }$ - $3 w$

Determine whether the following is always true or not always true. Given reasons for your answers. -(u × v) ·v = 0