Answer
$a.\qquad {\bf u}\cdot{\bf v}=3,\ |{\bf u}|=13,\ |{\bf v}|=1$
$b.\qquad 3/13$
$c.\qquad 3$
$d.\displaystyle \qquad \frac{9}{5}{\bf i}+ \frac{12}{5}{\bf k}$
Work Step by Step
${\bf u}=\langle 5, 12, 0\rangle \quad {\bf v}=\langle 3/5, 0, 4/5\rangle$
$a.$
${\bf u}\cdot{\bf v}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}=$
$=(5)(\displaystyle \frac{3}{5})+(12)(0)+(0)(4/5)$
$=3$
$|{\bf u}|=\sqrt{(5)^{2}+(12)^{2}+(0)^{2}}=\sqrt{25+144}=13$
$|{\bf v}|=\sqrt{(3/5)^{2}+(0)^{2}+(4/5)^{2}}=\sqrt{\frac{9+16}{25}}=1$
$b.$
$\displaystyle \cos\theta=\frac{{\bf u}\cdot{\bf v}}{|{\bf u}||{\bf v}|}=\frac{3}{(1)(13)}=\frac{3}{13}$
$c.$
$|{\bf u}|\displaystyle \cos\theta=13(\frac{3}{13})=3$
$d.$
$\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}){\bf v}$
$=\displaystyle \frac{3}{1^{2}}\langle 3/5, 0, 4/5\rangle$
$=\langle 9/5, 0, 12/5\rangle$
$= \displaystyle \frac{9}{5}{\bf i}+ \frac{12}{5}{\bf k}$
