Problem

2.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively. Position vectors are given relative to a fixed origin.]

Two particles, $A$ and $B$, are moving on a smooth horizontal surface. Each particle is moving with constant velocity.

At time $t$ seconds, the position vector of $A$ is given by $\mathbf{r}$ metres.

• When $t=2$, $\mathbf{r}=(-5\mathbf{i}+16\mathbf{j})$
• When $t=5$, $\mathbf{r}=(10\mathbf{i}+4\mathbf{j})$

(a) Find, in terms of $\mathbf{i}$ and $\mathbf{j}$, the velocity of $A$.

(b) Find an expression for $\mathbf{r}$ at time $t$ seconds. Give your answer in the form $p\mathbf{i}+q\mathbf{j}$, where $p$ and $q$ are functions of $t$

At time $t$ seconds, the position vector of $B$ is given by $\mathbf{s}$ metres where

$$\mathbf{s}=-5\mathbf{i}+7\mathbf{j}+t(2\mathbf{i}-3\mathbf{j})$$

(c) Find, to the nearest degree, the bearing of $B$ from $A$ when $t=5$