Edexcel IGCSE Math B 8.I vector geometry

Practise vector-geometry questions by writing routes in terms of a and b, comparing coefficients and using ratios on line segments.

Exam points

Write position or route vectors such as OP in terms of a and b from diagrams.
Use vector methods to find ratios such as OP:PD or points on a line segment.
Combine vector geometry with length or area information when the diagram requires it.

[Maximum number: 5]

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

(a)

Q is the point on OB such that OQ : OB = 1 : m where m is a constant.
$\overrightarrow{PQ}=\alpha\overrightarrow{AB}$ , where $\alpha$ is a scalar.

Using vectors, find the value of m and the value of $\alpha$ .

[ 3 ]

$\overrightarrow{PQ}=-\mathbf a+\frac{8}{m}\mathbf b$ $\overrightarrow{AB}=8\mathbf b-4\mathbf a$ $-\mathbf a+\frac{8}{m}\mathbf b=\alpha(8\mathbf b-4\mathbf a)$

Comparing coefficients:
$\alpha=\frac14,\quad m=4$

Marks: M1, A1, A1

(b)

Given that PR is parallel to OB,

find the value of n.

[ 2 ]

For $\overrightarrow{PR}$ to be parallel to $\overrightarrow{OB}$ , the coefficient of $\mathbf{a}$ in $\overrightarrow{PR}$ is zero:
$3-\frac{4}{n}=0$ $3=\frac{4}{n}$ $n=\frac{4}{3}$

OR, since $\overrightarrow{PR}=6\mathbf{b}$ ,
$\frac{8}{n}=6$ $n=\frac{4}{3}$

M1.
A1.

All question bank results loaded

8.I Vector geometry question 1

Question (a)

Question (b)