Question

Find the point of intersection of the following pair of lines, assuming that the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are not parallel.

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

• A. $2\overrightarrow{b}-2\overrightarrow{a}$
• B. $3\overrightarrow{b}-3\overrightarrow{a}$
• C. $2\overrightarrow{b}+4\overrightarrow{a}$
• D. $2\overrightarrow{a}-2\overrightarrow{b}$

Answer 1

Answer: (A)

$2\overrightarrow{b}-2\overrightarrow{a}$

Given,

$\overrightarrow{r}=\gamma (\overrightarrow{b}-\overrightarrow{a}),$------(1)

$\overrightarrow{r}=2\overrightarrow{b}+\mu \overrightarrow{a}.$------(2)

Equate two equations and find $\gamma ,\mu$
$\gamma (\overrightarrow{b}-\overrightarrow{a})=2\overrightarrow{b}+\mu \overrightarrow{a}$$\Rightarrow (\gamma -2)\overrightarrow{b}-(\gamma +\mu )\overrightarrow{a}=0$

On comparing we get,

$\Rightarrow \gamma -2=0,\gamma +\mu =0$
$\Rightarrow \gamma =2,\mu =-2$

putting value of $\gamma$ in eq(1) we get,

$\Rightarrow$ Point of intersection is $2b-2a$