Exam 13: Vectors and the Geometry of Space

Find the length and direction (when defined) of u × v. - $\mathbf { u } = - \frac { 1 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \mathbf { k } , \mathbf { v } = \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$

Find parametric equations for the line described below. -The line through the point P(5, -7, -4) and parallel to the line x = 5t - 4, y = 3t - 1, z = 2t + 3

Find v · u. - $\mathbf { v } = \left\langle \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right\rangle$ and $\mathbf { u } = \left( \frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 2 } } \right)$

Sketch the given surface. - $y = x ^ { 2 }$

Give a geometric description of the set of points whose coordinates satisfy the given conditions. - $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49 , z = 1$

Describe the given set of points with a single equation or with a pair of equations. -The circle of radius 2 centered at the point (10, -2, 2) and lying in a plane perpendicular to the x-axis

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

Express the vector as a product of its length and direction. -7j

Find an equation for the sphere with the given center and radius. -Center (-5, 7, 0), radius = 8

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. --5x - 3y - 8z = 7 and 6x + 5y - 9z = 8

Describe the given set of points with a single equation or with a pair of equations. -The plane through the point (-7, -3, 2) and parallel to the xy-plane

Find the vector projv u. -v = 2i - 2j - 4k, u = 5i - 12k

$\text { Use the vectors } u , v , w \text {, and } z \text { head to tail as needed to sketch the indicated vector. }$ - $2 \mathbf { u } - \mathbf { z } - \mathbf { w }$

Solve the problem. -Find the volume of the solid bounded by the ellipsoid $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$ and the planes $z = - 2$ and $z = 2$ . (The area of an ellipse with semiaxes a and $b$ is $\pi a b$ .)

Express the vector as a product of its length and direction. - $- 3 i - \frac { 5 } { 4 } j$

Find the magnitude. -Let $\mathbf { u } = \langle 3,4 \rangle$ . Find the magnitude (length) of the vector: $- 3 \mathbf { u }$ .

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 5 } + \frac { z ^ { 2 } } { 7 } = \frac { y } { 5 }$

Solve the problem. -Find a vector of magnitude 9 in the direction opposite to the direction of $\mathbf { v } = \frac { 1 } { 3 } \mathbf { i } + \frac { 1 } { 3 } \mathbf { j } - \frac { 1 } { 3 } \mathbf { k }$ .

Solve the problem. -Let u = 6i + j, v = i + j, and w = i - j. Find scalars a and b such that u = av + bw.

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (-8, -3); parallel to v = -3i + 7j