Practice section 4 of 9

$\overrightarrow{OA}=\mathbf i+4\mathbf j+7\mathbf k$ makes angles $\theta,\phi,\omega$ with the positive x-, y-, z-axes respectively. Show that $\cos^2\theta+\cos^2\phi+\cos^2\omega=1$.

Given $\mathbf a=\begin{pmatrix}1\\2\\2\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}2\\1\\2\end{pmatrix}$, find (i) the length of projection of $\mathbf a$ on $\mathbf b$; (ii) the projection of $\mathbf a$ on $\mathbf b$; (iii) the component of $\mathbf a$ perpendicular to $\mathbf b$.

Find the projection of $\mathbf a=\begin{pmatrix}-1\\2\\-2\end{pmatrix}$ on $\mathbf b=\begin{pmatrix}2\\1\\2\end{pmatrix}$.

Given $\mathbf a\times\mathbf b=\mathbf c\times\mathbf a$, $|\mathbf a|\ne0$, and nonparallel vectors $\mathbf b,\mathbf c$, show that $\mathbf a$ is parallel to $\mathbf b+\mathbf c$.

Find a vector perpendicular to $\mathbf a=\begin{pmatrix}2\\1\\1\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}2\\1\\4\end{pmatrix}$.

Find the vector product of (i) $\begin{pmatrix}1\\1\\7\end{pmatrix}$ and $\begin{pmatrix}5\\-4\\2\end{pmatrix}$; (ii) $2\mathbf i+4\mathbf j+4\mathbf k$ and $\mathbf i+\mathbf j-\mathbf k$.