Answer
$a.\qquad {\bf u}\cdot{\bf v}=10+\sqrt{17},\ |{\bf u}|=\sqrt{21},\ |{\bf v}|=\sqrt{26}$
$b.\displaystyle \qquad \frac{10+\sqrt{17}}{\sqrt{546}}$
$c.\displaystyle \qquad \frac{10+\sqrt{17}}{\sqrt{26}}$
$d.\displaystyle \qquad \frac{5(10+\sqrt{17})}{26}{\bf i} + \frac{10+\sqrt{17}}{26}{\bf j}$
Work Step by Step
${\bf u}=\langle 2,\sqrt{17},0\rangle \quad {\bf v}=\langle 5,1,0\rangle$
${\bf (a)}$
${\bf u}\cdot{\bf v}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}=$
$=(2)(5)+(\sqrt{17})(1)+(0)(0)$
$=10+\sqrt{17}$
$|{\bf u}|=\sqrt{(2)^{2}+(\sqrt{17})^{2}+(0)^{2}}=\sqrt{21}$
$|{\bf v}|=\sqrt{(5)^{2}+(1)^{2}+(0)^{2}}=\sqrt{26}$
${\bf (b)}$
$\displaystyle \cos\theta=\frac{{\bf u}\cdot{\bf v}}{|{\bf u}||{\bf v}|}=\frac{10+\sqrt{17}}{(\sqrt{21})(\sqrt{26})}=\frac{10+\sqrt{17}}{\sqrt{546}}$
${\bf (c)}$
$|{\bf u}|\displaystyle \cos\theta=\sqrt{21}(\frac{10+\sqrt{17}}{(\sqrt{21})(\sqrt{26})})=\frac{10+\sqrt{17}}{\sqrt{26}}$
${\bf (d)}$
$\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}){\bf v}$
$=\displaystyle \frac{10+\sqrt{17}}{26}\langle 5,1,0\rangle$
$= \displaystyle \frac{5(10+\sqrt{17})}{26}{\bf i} + \frac{10+\sqrt{17}}{26}{\bf j}$
