CAIE A-Level Mathematics A2 3.7.6 Scalar Product Questions

Practise scalar products, angles and perpendicularity in vector geometry.

Exam points

calculate scalar products of vectors
use scalar products to find angles and perpendicularity
apply scalar products to solve vector geometry problems

[Maximum number: 3]

The equations of two straight lines $l_{1}$ and $l_{2}$ are

l_{1}: \quad \mathbf{r}=\mathbf{i}-2 \mathbf{j}+3 \mathbf{k}+\lambda(2 \mathbf{i}-\mathbf{j}+a \mathbf{k}) \quad \text { and } \quad l_{2}: \quad \mathbf{r}=-\mathbf{i}-\mathbf{j}-\mathbf{k}+\mu(3 \mathbf{i}-2 \mathbf{j}-2 \mathbf{k}),

where a is a constant.
The lines $l_{1}$ and $l_{2}$ are perpendicular.

(a)

Show that a=4.

[ 1 ]

(b)

The point B is the image of A after a reflection in the line $l_{2}$ .

Find the position vector of B.

[ 2 ]

Method to find position vector of B
E.g. $\pm 2 \times$ their $(-\mathbf{i}-\mathbf{j}-\mathbf{k}) \pm(-5 \mathbf{i}+\mathbf{j}-9 \mathbf{k})$

Obtain position vector of B is 3 i-3 j+7 k

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CAIE A-Level Mathematics A2 3.7.6 Scalar Product Questions question 1

Question (a)

Question (b)