Exam 13: A Fundamental Tool- Vectors

Suppose $\vec { u } = \vec { i } + 4 \vec { j } + 5 \vec { k }$ and $\vec { v }$ is a vector such that $\vec { u } \cdot \vec { v } = 37$ .What is the shortest length $\vec { v }$ can be?

True or false: The vector $- \frac { \vec { a } } { \| \vec { a } \| }$ is a unit vector provided $\| \vec { a } \| \neq 0$ .Give a reason for your answer.

If $\vec { w } = \vec { i } - 2 \vec { j } + \vec { k }$ , find a unit vector parallel to $\bar{W}$

Consider the triangle ABC where A is the point (4, 5, -3), B is the point (0, -3, 2)and C is the point (4, 1, 1). Find the cosine of angle BAC.Give your answer to 4 decimal places.

Choose values of b and c so that the following planes do not intersect. $2 x + b y - c z = 24$ and $2 x - 4 y = 25$

The two planes z = -3x - 4y + 5 and -6x - 8y - 2z = 0 are parallel.

Suppose $\vec { u } = 4 \vec { i } + b \vec { j } + 3 \vec { k }$ and $\vec { v } = a \vec { i } + 6 \vec { j } + 3 \vec { k }$ Find values for a and b making $\vec { u }$ perpendicular to both $\vec { v }$ and $\bar { j }$