Vectors in two dimensions

Find $\overrightarrow{OD}$.

Hence find the value of $\lambda$.

Given that $\overrightarrow{PQ} = \begin{pmatrix} -3 \\ 7 \end{pmatrix}$ and $4\overrightarrow{PR} = \begin{pmatrix} -2 \\ 8 \end{pmatrix}$, find $\overrightarrow{RQ}$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a} = \alpha \mathbf{i} + 6\mathbf{j}$, $\mathbf{b} = 4\mathbf{i} + \beta \mathbf{j}$ and $\mathbf{c} = (2\alpha + 5\beta)\mathbf{i} + 20\mathbf{j}$, where $\alpha$ and $\beta$ are scalars.

Given that $\mathbf{c} = 3\mathbf{a} - 2\mathbf{b}$, find the values of $\alpha$ and $\beta$.

Find $\overrightarrow{OR}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mu$.

Hence find the values of $\lambda$ and $\mu$.

The point $R$ is such that $\overrightarrow{OR}$ is in the same direction as $\overrightarrow{PQ}$ and the magnitude of $\overrightarrow{OR}$ is $3\sqrt{26}$.

Find $\overrightarrow{OR}$.

$\overrightarrow{OP}$ is in the same direction as $2\mathbf{i} - 3\mathbf{j}$ and $\overrightarrow{OQ} = 10\mathbf{i} + 6\mathbf{j}$.

Find $\overrightarrow{OP}$.

Find the velocity vector of $P$.

Initially, $P$ has position vector $\begin{pmatrix}3 \\ 5\end{pmatrix}$.

Write down the position vector of $P$ at time $t$.

A second particle $Q$ has position vector $\begin{pmatrix}-1 \\ 3\end{pmatrix} + \begin{pmatrix}-5 \\ 7.5\end{pmatrix}t$ at time $t$.

Find, in terms of $t$, the distance between $P$ and $Q$ at time $t$. Simplify your answer.

Hence show that $P$ and $Q$ never collide.

$\overrightarrow{AB}$

$\overrightarrow{OD}$.

Find $\overrightarrow{OX}$ in terms of $\mathbf{a}$, $\mathbf{c}$ and $\mu$.

Find $\overrightarrow{AX}$ in terms of $\mathbf{a}$, $\mathbf{c}$ and $\lambda$.

Hence find the values of $\lambda$ and $\mu$.

Find $\overrightarrow{OZ}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\lambda$.

Find $\overrightarrow{OZ}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mu$.

Find the values of $\lambda$ and $\mu$.

Hence find $\overrightarrow{OZ}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ only.

Show that $\overrightarrow{XY} = \frac{2}{3}\mathbf{a} - \frac{7}{15}\mathbf{b}$.

Find $\overrightarrow{YZ}$ in terms of $\lambda$, $\mathbf{a}$ and $\mathbf{b}$.

Find $\overrightarrow{YZ}$ in terms of $\mu$, $\mathbf{a}$ and $\mathbf{b}$.

Hence find the values of $\lambda$ and $\mu$.

Find the vector which is in the opposite direction to $\begin{pmatrix} 15 \\ -8 \end{pmatrix}$ and has a magnitude of 8.5.

Find the values of $a$ and $b$ such that