Vectors & vector geometry

$\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$

(a) Write $2\mathbf{a} + \mathbf{b}$ as a column vector. [1]

(b) Write $\mathbf{a} - 3\mathbf{b}$ as a column vector. [2]

$\mathbf{u} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 7 \end{pmatrix}$

Leila says, “The magnitude of $\mathbf{u} + \mathbf{v}$ is equal to the magnitude of $\mathbf{u}$ added to the magnitude of $\mathbf{v}$.”

Show that Leila is wrong. You must show your working. [3]

$A$ is the point $(2, 5)$ and $B$ is the point $(7, -1)$.

(a) Write $\overrightarrow{AB}$ as a column vector. [1]

$\overrightarrow{BC} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$

(b) Find the coordinates of $C$. [1]

(c) Work out $|\overrightarrow{AC}|$. Give your answer correct to 3 significant figures. [2]

$\mathbf{p} = \begin{pmatrix} 6 \\ -9 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}$

(a) Explain why $\mathbf{p}$ and $\mathbf{q}$ are parallel. [2]

(b) Find the ratio $|\mathbf{p}| : |\mathbf{q}|$. [1]

$m\begin{pmatrix} 2 \\ 3 \end{pmatrix} + n\begin{pmatrix} 1 \\ -4 \end{pmatrix} = \begin{pmatrix} 8 \\ 1 \end{pmatrix}$

where $m$ and $n$ are numbers.

Find the value of $m$ and the value of $n$. [3]

$OAB$ is a triangle.

$\overrightarrow{OA} = \mathbf{a}$    $\overrightarrow{OB} = \mathbf{b}$

$P$ is the point on $AB$ such that $AP : PB = 2 : 3$

Find $\overrightarrow{OP}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Give your answer in its simplest form. [3]

$ABCDEF$ is a regular hexagon with centre $O$.

$\overrightarrow{OA} = \mathbf{a}$    $\overrightarrow{OB} = \mathbf{b}$

(a) Find, in terms of $\mathbf{a}$ and $\mathbf{b}$,  (i) $\overrightarrow{AB}$   (ii) $\overrightarrow{OC}$ [2]

(b) Find $\overrightarrow{BF}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Give your answer in its simplest form. [2]

$OAB$ is a triangle.

$\overrightarrow{OA} = 3\mathbf{a}$    $\overrightarrow{OB} = 3\mathbf{b}$

$P$ is the point on $AB$ such that $AP : PB = 2 : 1$

$\overrightarrow{BR} = 2\mathbf{a} + \mathbf{b}$

Use a vector method to prove that $O$, $P$ and $R$ lie on a straight line. [5]

$OAB$ is a triangle.

$\overrightarrow{OA} = \mathbf{a}$    $\overrightarrow{OB} = \mathbf{b}$

$P$ is the point on $OA$ such that $OP : PA = 1 : 2$
$Q$ is the midpoint of $OB$.

The lines $AQ$ and $BP$ intersect at the point $X$.

Find $\overrightarrow{OX}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Show your working clearly. [5]

A drone makes three straight flights, one after the other. The displacement of each flight, in metres, is

$\begin{pmatrix} 120 \\ 50 \end{pmatrix}$,   $\begin{pmatrix} -30 \\ 90 \end{pmatrix}$  and  $\begin{pmatrix} -40 \\ -160 \end{pmatrix}$

where the top number is the distance east and the bottom number is the distance north.

(a) Find the resultant displacement of the drone as a column vector. [2]

(b) Work out the straight-line distance of the drone from its starting point at the end of the three flights. Give your answer correct to 3 significant figures. [2]