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