Vectors in two dimensions

Find the position vector of $A$ at the end of 2 hours of sailing.

Write down the position vector of $A$ at the end of $t$ hours of sailing.

Point $Q$ has position vector $20\mathbf{i} + 8\mathbf{j}$ relative to the origin $O$.
At 1500, boat $B$ sails from point $Q$ on a bearing of $300^\circ$ with a constant speed of $5\sqrt{3}$.

Find the position vector of $B$ at 1700.

Find the distance between the two boats at 1700.

A particle $P$ is moving in a straight line with a speed of $26$ in the direction of the vector $\begin{pmatrix}5 \\ -12\end{pmatrix}$.

Find the velocity vector of $P$.

When $t = 0$, $P$ passes through a point $A$ which has position vector $\begin{pmatrix}3 \\ 6\end{pmatrix}$.

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

At the same time that $P$ passes through $A$, a particle $Q$ passes through a point $B$.

The position vector of $Q$ at time $t$ is given by $\begin{pmatrix}8t - 5 \\ 2 - 25t\end{pmatrix}$.

The distance between $P$ and $Q$ at time $t$ is $d$.

Show that $d^2 = mt^2 + nt + r$, where $m$, $n$ and $r$ are integers to be found.

Hence show that $P$ and $Q$ do not collide.

Find two expressions for $\overrightarrow{OP}$, each in terms of $\mathbf{b}$, $\mathbf{c}$ and a scalar, and hence show that $P$ divides both $AC$ and $DB$ in the ratio $4 : 7$.

The point $Q$ is such that $\overrightarrow{OQ} = \frac{2}{7}\mathbf{b} + \frac{2}{7}\mathbf{c}$.

Use a vector method to show that $O$, $Q$ and $P$ are collinear. Justify your answer.

Find the exact values of the components of the velocity of particle $A$ in the $x$-direction and the $y$-direction.

Find, in terms of $t$, the position vector of particle $A$ at time $t$.

Particle $B$ starts from the point $(2\sqrt{3}, 9)$ at time $t = 0$. It moves with constant speed $\frac{5}{3}\,\text{ms}^{-1}$ parallel to the positive $x$-axis.

Find, in terms of $t$, the position vector of particle $B$ at time $t$.

Hence show that the particles collide.

The position vectors of the points $P$, $Q$ and $R$ relative to an origin $O$ are $\begin{pmatrix} 4 \\ 7 \end{pmatrix}$, $\begin{pmatrix} 8 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} x \\ y \end{pmatrix}$ respectively. The point $R$ lies on $PQ$ extended such that $3\overrightarrow{QR} = 2\overrightarrow{PR}$. Use a vector method to find the values of $x$ and $y$.

Find $\mathbf{a}$ and $\mathbf{c}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.

Find the magnitude and bearing of $\mathbf{b}$.

Show that the velocity vector of $A$ is $9\mathbf{i} - 3\sqrt{3}\mathbf{j}$.

Find the position vector of $A$ at 1200.

At 11 00 ship $B$ leaves a point $Q$ with position vector $29\mathbf{i} + 16\mathbf{j}$. It sails with constant velocity $-12\sqrt{3}\mathbf{j}$. Write down the position vector of $B$, $t$ hours after it starts sailing.

Find the distance between the two ships at 1200.

In this question, $\mathbf{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north.

A cyclist rides at a speed of $4\,\text{ms}^{-1}$ on a bearing of $015^\circ$. Write the velocity vector of the cyclist in the form $x\mathbf{i} + y\mathbf{j}$, where $x$ and $y$ are constants.

A vector of magnitude $6$ on a bearing of $300^\circ$ is added to a vector of magnitude $2$ on a bearing of $230^\circ$ to give a vector $\mathbf{v}$. Find the magnitude and bearing of $\mathbf{v}$.

Particle $A$ starts from the point with position vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and travels with speed $26\,\text{ms}^{-1}$ in the direction of the vector $\begin{pmatrix} 12 \\ 5 \end{pmatrix}$. Find the position vector of $A$ after $t$ seconds.

At the same time, particle $B$ starts from the point with position vector $\begin{pmatrix} 67 \\ -18 \end{pmatrix}$. It travels with speed $20\,\text{ms}^{-1}$ at an angle of $\alpha$ above the positive $x$-axis, where $\tan\alpha = \frac{3}{4}$. Find the position vector of $B$ after $t$ seconds.

Hence find the time at which $A$ and $B$ meet, and the position where this occurs.

Given that $\overrightarrow{PX} = \lambda\overrightarrow{PS}$, write $\overrightarrow{OX}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\lambda$.

Given that $\overrightarrow{OX} = \mu\overrightarrow{OQ}$, write $\overrightarrow{OX}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mu$.

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

Write down the value of $\frac{OX}{OQ}$.

Find the value of $\frac{PX}{XS}$.