IGCSE Math B 8.E Vector sums and differences Topic Practice

Given that $\mathbf{a}=\binom{2}{3}$ and $\mathbf{b}=\binom{1}{-2}$
find, as a column vector, a-2 b

Figure 2 shows the triangle OAB in which $\overrightarrow{O A}=6 \mathbf{a}$ and $\overrightarrow{O B}=4 \mathbf{b}$
The point C lies on OA such that O C: O A=2: 3

Find, in terms of a and b or a or b, simplifying your answer where possible,

$\overrightarrow{AB}$

Figure 1 shows triangle OAB in which $\overrightarrow{O A}=12 \mathbf{a}$ and $\overrightarrow{O B}=8 \mathbf{b}$.

Express $\overrightarrow{AB}$ in terms of $\mathbf a$ and $\mathbf b$.

Figure 1 shows triangle OAB in which $\overrightarrow{O A}=4 \mathbf{a}$ and $\overrightarrow{O B}=8 \mathbf{b}$
P is the point on OA such that O P: O A=1: 4

$21 \overrightarrow{O X}=\binom{2}{5}$ and $\overrightarrow{O Y}=\binom{-2}{7}$

Express $\overrightarrow{X Y}$ as a column vector.

The vectors
$$\mathbf p=\begin{pmatrix}2x-1\\y\end{pmatrix}, \qquad \mathbf q=\begin{pmatrix}y+3\\-y\end{pmatrix}$$
are such that $|\mathbf p|=\sqrt{98}$ and
$$\mathbf p+\mathbf q=\begin{pmatrix}7\\0\end{pmatrix}.$$

Show that
$$y=2-3\sqrt5.$$

Diagram NOT accurately drawn

Figure 4 shows quadrilateral OABC with CB parallel to OA.
$\overrightarrow{O A}=12 \mathbf{a} \quad \overrightarrow{O B}=6 \mathbf{b}$ and $C B=\frac{1}{2} O A$

Figure 2 shows triangle OAB with $\overrightarrow{OA}=4\mathbf a$ and $\overrightarrow{OB}=3\mathbf b$.
The point C lies on OB such that $\overrightarrow{OC}=2\mathbf b$.
The point D is the midpoint of AB.
The point E lies on AC such that OED is a straight line.

Figure 3 shows three points A, B and C on a circle with centre O where AOB is a diameter of the circle.
D is the point on AC such that A D: D C=3: 2
Given that $\overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{A D}=\mathbf{b}$

Figure 4 shows a trapezium OACB in which $\angle O A C=\angle A O B=90^{\circ}$ so that OB and AC are parallel.
Given that $\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}$ and $\overrightarrow{A C}=2 \mathbf{b}$,