Q1
Answer
The points $A$ and $B$ have position vectors $$\mathbf{r}_A = \mathbf{i} + \mathbf{j} + \mathbf{k}$$ $$\mathbf{r}_B = 3\mathbf{i} - \mathbf{j} - \mathbf{k}$$ Find $\overrightarrow{AB}$.
$2\mathbf{i} - 2\mathbf{j} - 2\mathbf{k}$
Q2
Answer
The points $A$ and $B$ have position vectors $$\mathbf{r}_A = 2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$$ $$\mathbf{r}_B = 5\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$$ Find $|\overrightarrow{AB}|$.
$\sqrt{74}$
Q3
Answer
The points $A$ and $B$ have position vectors $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ and $\mathbf{b} = 5\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ respectively.
  1. Find the vector $\overrightarrow{AB}$.
    1. Find the unit vector in the direction of $\overrightarrow{AB}$.
      1. Find the position vector of the midpoint $M$ of $AB$.
        (a) $3\mathbf{i} - 4\mathbf{j} + 3\mathbf{k}$ $\quad$ (b) $\dfrac{1}{\sqrt{34}}(3\mathbf{i} - 4\mathbf{j} + 3\mathbf{k})$ $\quad$ (c) $\dfrac{7}{2}\mathbf{i} + \mathbf{j} + \dfrac{1}{2}\mathbf{k}$
        1. $\overrightarrow{AB} = 3\mathbf{i} - 4\mathbf{j} + 3\mathbf{k}$
        2. $\hat{u} = \dfrac{1}{\sqrt{34}}(3\mathbf{i} - 4\mathbf{j} + 3\mathbf{k})$
        3. $\mathbf{m} = \dfrac{7}{2}\mathbf{i} + \mathbf{j} + \dfrac{1}{2}\mathbf{k}$
        Q4
        Answer
        The points $A$, $B$ and $C$ have position vectors $$\mathbf{r}_A = 3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$$ $$\mathbf{r}_B = \mathbf{i} + \mathbf{j} + \mathbf{k}$$ $$\mathbf{r}_C = 5\mathbf{i} + 7\mathbf{j} - 3\mathbf{k}$$ Determine if the poitns are collinear.
        $\overrightarrow{AB} = -\overrightarrow{AC}$ and they share the point $A$, so yes.
        Q5
        Answer
        The points $A$, $B$ and $C$ are collinear and have position vectors $$\mathbf{r}_A = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$$ $$\mathbf{r}_B = 3\mathbf{i} - \mathbf{j} + \mathbf{k}$$ $$\mathbf{r}_C = 5\mathbf{i} + p\mathbf{j} + q\mathbf{k}$$ Find $p$ and $q$.
        $p = -4$ and $q = 5$
        Q6
        Answer
        The points $A$ and $B$ have position vectors $$\mathbf{r}_A = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$$ $$\mathbf{r}_B = \mathbf{i} + \mathbf{j} - \mathbf{k}$$ Show that the triangle $OAB$ is right angled.
        Pythagoras
        Q7
        Answer
        The points $A$ $B$ and $C$ have position vectors $$\mathbf{r}_A = -6\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}$$ $$\mathbf{r}_B = -3\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$$ $$\mathbf{r}_C = -8\mathbf{i} + 1\mathbf{j} - 6\mathbf{k}$$ Show that the triangle $ABC$ is right angled.
        Pythagoras
        Q8
        Answer
        The points $A$, $B$ and $C$ have position vectors $$\mathbf{r}_A = \mathbf{i} + \mathbf{j} - \mathbf{k}$$ $$\mathbf{r}_B = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$$ $$\mathbf{r}_C = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$$ For the triangle $ABC$, find, to 3 significant figures,
        1. the perimeter
        2. the area.
        1. $13.2$
        2. $2.55$
        Q9
        Answer
        The points $A$, $B$ and $C$ have position vectors $$\mathbf{r}_A = 4\mathbf{i} - \mathbf{j} - 2\mathbf{k}$$ $$\mathbf{r}_B = \mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$$ $$\mathbf{r}_C = -3\mathbf{i} + 2\mathbf{j} + \mathbf{k}$$ For the triangle $ABC$, find, to 3 significant figures,
        1. the perimeter
        2. the area.
        1. $21.3$
        2. $20.6$
        Q10
        Answer
        The points $A$ $B$ and $C$ have position vectors $$\mathbf{r}_A = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$$ $$\mathbf{r}_B = 5\mathbf{i} + 2\mathbf{j} + 7\mathbf{k}$$ $$\mathbf{r}_C = \mathbf{i} + 4\mathbf{j} + 6\mathbf{k}$$ Given $ABCD$ is a parallelogram, find the position vector of $D$.
        $-2\mathbf{i} + \mathbf{j} + 2\mathbf{k}$
        Q11
        Answer
        The point $A$ has the two dimensional position vector $\mathbf{r} = 6\mathbf{i}$. Given $OAB$ is an equilateral triangle, find the possible position vectors for $B$.
        $3\mathbf{i} + 3\sqrt{3}\mathbf{j}$ or $3\mathbf{i} - 3\sqrt{3}\mathbf{j}$
        Q12
        Answer
        The forces $\mathbf{F}_1$ and $\mathbf{F}_2$ act on a particle. $$\mathbf{F}_1 = 2\mathbf{i} - \mathbf{j}$$ $$\mathbf{F}_2 = -3\mathbf{i} + 5\mathbf{j}$$ Find the magnitude and direction of the resultant force.
        $\sqrt{17}$, $104^{\circ}$
        Q13
        Answer
        The forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ act on a particle. $$\mathbf{F}_1 = 4\mathbf{i} - 2\mathbf{j}$$ $$\mathbf{F}_2 = 2\mathbf{i} + p\mathbf{j}$$ $$\mathbf{F}_3 = q\mathbf{i} - 3\mathbf{j}$$ The particle is in equilibrium. Find $p$ and $q$.
        $p = 5$ and $q = -6$
        Q14
        Answer
        The forces $\mathbf{F}_1$ and $\mathbf{F}_2$ act on a particle Given $\mathbf{F}_1 = 5\mathbf{i} - 3\mathbf{j}$ and that the resultant force has magnitude $\sqrt{8}$ and direction $-45^{\circ}$, find $\mathbf{F}_2$.
        $-3\mathbf{i} + \mathbf{j}$
        Q15
        Answer
        The forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ act on a particle. $$\mathbf{F}_1 = \mathbf{j}$$ $$\mathbf{F}_2 = 2\mathbf{i} + 3\mathbf{j}$$ $$\mathbf{F}_3 = p\mathbf{i} + q\mathbf{j}$$ The resultant force has magnitude $\sqrt{18}$ and direction $135^{\circ}$. Find $p$ and $q$.
        $p = -5$ and $q = -1$
        Q16
        Answer
        A particle of mass 500 g is being acted on by a force $\mathbf{F} = \mathbf{i} + \mathbf{j}$. Find the acceleration of the particle.
        $2\mathbf{i} + 2\mathbf{j}$
        Q17
        Answer
        A particle of mass 2 kg is being acted on by a force $\mathbf{F} = -4\mathbf{i} + 2\mathbf{j}$. Find the magnitude and direction of the acceleration of the particle.
        $\sqrt{5}$, $153^{\circ}$