In the figure, a vector $\overrightarrow{a}$ with a magnitude of $15.0 \mathrm{m}$ is directed at angle $\mathrm{\theta }=56.0°$ counter-clockwise from the $+x$-axis. What are the components (a) $a_{\mathrm{x}}$ and (b) $a_y$ of the vector? A second coordinate system is inclined by an angle $\mathrm{\theta }\text{'}=18.0°$ with respect to the first. What are the components (c) $a{\text{'}}_{\mathrm{x}}$ and (d) $a{\text{'}}_{\mathrm{y}}$ in this primed coordinate system?

   

For the vectors in the figure, with $a=4, b=3$ and $c=5$, what are

(a) the magnitude and (b) the direction of $\overrightarrow{a}\times \overrightarrow{b}$ and

(c) the magnitude and (d) the direction of $\overrightarrow{a}\times \overrightarrow{c}$ and

(e) the magnitude and (f) the direction of $\overrightarrow{b}\times \overrightarrow{c}$? (The $z-\mathrm{axis}$ is not shown)

In the figure, a cube of edge length $\mathrm{a}$ sits with one corner at the origin of an $\mathrm{xyz}$ coordinate system. A body diagonal is a line that extends from one corner to another through the center. In unit-vector notation, what is the body diagonal that extends from the corner at:

(a) coordinates $\left(0, 0\right)$,

(b) coordinates $\left(\mathrm{a}, 0, 0\right)$,

(c) coordinates $\left(0, a, 0\right)$,

(d) co-ordinates $\left(a, a, 0\right)$,

(e) Determine the angles that the body diagonals make with the adjacent edges.

(f) Determine the length of the body diagonals in terms of $\mathrm{a}$.

For the following three vectors, what is $3\overrightarrow{C}\cdot \left(2\overrightarrow{A}\times \overrightarrow{B}\right)$?

$\overrightarrow{A}=2.00\hat{\mathrm{i}}+3.00\hat{\mathrm{j}}-4.00\hat{\mathrm{k}}$

$\overrightarrow{B}=-3.00\hat{\mathrm{i}}+4.00\hat{\mathrm{j}}+2.00\hat{\mathrm{k}}$

$\overrightarrow{C}=7.00\hat{\mathrm{i}}-8.00\hat{\mathrm{j}}$