Exam 13: Vectors and the Geometry of Space

Solve the problem. -Find the volume of the solid bounded by the elliptical cone $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 81 } = \frac { z ^ { 2 } } { 16 }$ and the planes $z = 0$ and $z = 4$ . (The area of an ellipse with semiaxes $a$ and $b$ is $\pi a b$ .)

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

Find the angle between u and v in radians. -u = 8i, v = 6i - 4j

Solve the problem. -A garden hose is spraying a stream of water at a box on the ground with a force of 2.4 pounds. The water stream makes an angle of 30 degrees with the ground. What is the horizontal component of the force?

Calculate the requested distance. -The distance from the point S(-1, 9, 8) to the line x = -4 + 2t, y = -2 + 11t, z = 7 + 10t

Find the component form of the specified vector. -The unit vector that makes an angle $\pi / 3$ with the positive $x$ -axis

Find the intersection. -x = -10 - 10t, y = + 4t, z = -6 + 4t ; x - 10y - 8z = -3

Find the triple scalar product (u x v) · w of the given vectors. -u = -5i - 3j + 4k; v = 8i - 6j + 7k; w = 8i - 7j - 10k

Find the indicated vector. -Let $\mathbf { u } = \langle - 8 , - 5 \rangle , \mathbf { v } = \langle 4,9 \rangle$ . Find $\mathbf { u } + \mathbf { v }$ .