Answer
$a.\qquad {\bf u}\cdot{\bf v}=2,\ |{\bf u}|=\sqrt{3},\ |{\bf v}|=\sqrt{34}$
$b.\displaystyle \qquad \frac{\sqrt{102}}{51}$
$c.\displaystyle \qquad \frac{\sqrt{34}}{17}$
$d.\displaystyle \qquad \frac{5}{17}{\bf j} -\frac{3}{17}{\bf k}$
Work Step by Step
${\bf u}=\langle 1,1,1\rangle \quad {\bf v}=\langle 0,5,-3\rangle$
${\bf (a)}$
${\bf u}\cdot{\bf v}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}=$
$=(1)(0)+(1)(5)+(1)(-3)$
$=5-3$
$=2$
$|{\bf u}|=\sqrt{(1)^{2}+(1)^{2}+(1)^{2}}=\sqrt{3}$
$|{\bf v}|=\sqrt{(0)^{2}+(5)^{2}+(-3)^{2}}=\sqrt{25+9}=\sqrt{34}$
${\bf (b)}$
$\displaystyle \cos\theta=\frac{{\bf u}\cdot{\bf v}}{|{\bf u}||{\bf v}|}=\frac{2}{(\sqrt{3})(\sqrt{34})}$
$=\displaystyle \frac{2}{\sqrt{102}}=\frac{2\sqrt{102}}{102} = \frac{\sqrt{102}}{51}$
${\bf (c)}$
$|{\bf u}|\displaystyle \cos\theta=\sqrt{3}(\frac{2}{(\sqrt{3})(\sqrt{34})})$
$=\displaystyle \frac{2}{\sqrt{34}}=\frac{2\sqrt{34}}{34}=\frac{\sqrt{34}}{17}$
${\bf (d)}$
$\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}){\bf v}$
$=\displaystyle \frac{2}{34}\langle 0,5,-3\rangle$
$=\displaystyle \langle 0,\frac{5}{17},-\frac{3}{17}\rangle$
$= \displaystyle \frac{5}{17}{\bf j} -\frac{3}{17}{\bf k}$
