Practice section 3 of 9

If $\mathbf a=\begin{pmatrix}1\\t\\2\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}0\\1\\2\end{pmatrix}$ are perpendicular, find $t$.

The angle between $\mathbf a$ and $\mathbf b$ is $\pi/3$ radians, and the angle between $\mathbf b$ and $\mathbf a-\mathbf b$ is a right angle. Show that $|\mathbf a|=2|\mathbf b|$.

Find the angle between $\mathbf a=\begin{pmatrix}1\\2\\2\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}1\\3\\-4\end{pmatrix}$.

Calculate the angle between $\mathbf a$ and $\mathbf b$ if (i) $\mathbf a=2\mathbf i-3\mathbf j+6\mathbf k$, $\mathbf b=\mathbf i+\mathbf j$; (ii) $\mathbf a=2\mathbf i-4\mathbf j+5\mathbf k$, $\mathbf b=\mathbf i+3\mathbf j+8\mathbf k$.

$A,B,C$ have position vectors $\mathbf i+2\mathbf j+3\mathbf k$, $\mathbf i+\mathbf j+\mathbf k$ and $2\mathbf i+3\mathbf j-2\mathbf k$ respectively. Find $\angle ABC$ to the nearest $0.1^\circ$.

Which of the following is/are correct?

(1) $\overrightarrow{PQ}\cdot\overrightarrow{QR}=|\overrightarrow{PQ}||\overrightarrow{QR}|\cos\angle PQR$.

(2) $\overrightarrow{BA}\cdot\overrightarrow{BC}=|\overrightarrow{BA}||\overrightarrow{BC}|\cos\angle ABC$.

(3) $\overrightarrow{KM}\cdot\overrightarrow{NM}=|\overrightarrow{KM}||\overrightarrow{NM}|\cos\angle KMN$.