Question

Find the points of intersection whenever they exist of the following pairs of lines:

$\overrightarrow{r}=3(a+c)+\lambda (b-c),\overrightarrow{r}=2(b-c)+\mu (a+b)$

• A. The lines intersect at the point $3a+5b-2c$
• B. The lines intersect at the point $3a+b+2c$
• C. The lines intersect at the point $a+3b-c$
• D. The lines intersect at the point $3a+2b+c$

Answer 1

The correct option is A The lines intersect at the point $3a+5b-2c$

To find point of intersection of two lines equate both lines and find $\lambda ,\mu$
Given $\overrightarrow{r}=3(a+c)+\lambda (b-c),\overrightarrow{r}=2(b-c)+\mu (a+b)$
$3(a+c)+\lambda (b-c)=2(b-c)+\mu (a+b)$$\Rightarrow (3-\mu )a+(3-\lambda +2)c+(\lambda -2-\mu )b=0$
$\Rightarrow 3-\mu =0,5-\lambda =0,\lambda -2-\mu =0$
$\Rightarrow \mu =3,\lambda =5$
Substitute these values in above equations, we get point of intersection $=3a+5b-2c$