Answer
$a.\qquad {\bf u}\cdot{\bf v}=25,\ |{\bf u}|=5,\ |{\bf v}|=15$
$b.\qquad 1/3$
$c.\qquad 5/3$
$d.\displaystyle \qquad \frac{10}{9}{\bf i}+ \frac{11}{9}{\bf j} - \frac{2}{9}{\bf k}$
Work Step by Step
${\bf u}=\langle 0, 3, 4\rangle \quad {\bf v}=\langle 10, 11, -2\rangle$
$a.$
${\bf u}\cdot{\bf v}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}=$
$=(0)(10)+(3)(11)+(4)(-2)$
$=33-8$
$=25$
$|{\bf u}|=\sqrt{(0)^{2}+(3)^{2}+(4)^{2}}=\sqrt{9+16}=5$
$|{\bf v}|=\sqrt{(10)^{2}+(11)^{2}+(-2)^{2}}=\sqrt{100+121+4}=15$
$b.$
$\displaystyle \cos\theta=\frac{{\bf u}\cdot{\bf v}}{|{\bf u}||{\bf v}|}=\frac{25}{(5)(15)}=\frac{1}{3}$
$c.$
$|{\bf u}|\displaystyle \cos\theta=5(\frac{1}{3})=\frac{5}{3}$
$d.$
$\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}){\bf v}$
$=\displaystyle \frac{25}{15^{2}}\langle 10, 11, -2\rangle$
$=\displaystyle \frac{1}{9}\langle 10, 11, -2\rangle$
$= \displaystyle \frac{10}{9}{\bf i}+ \frac{11}{9}{\bf j} - \frac{2}{9}{\bf k}$
