Question

If $\hat{a}and\hat{b}$ are non collinerar vectors, find the value of X for which the vectors $\overrightarrow{l}=(2x+1)\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{m}=(x-2)\overrightarrow{a}+\overrightarrow{b}$ are collinear.

Answer 1

Given $\overrightarrow{l}=(2x+1)\overrightarrow{a}-\overrightarrow{b}$

$\overrightarrow{m}=(x-2)\overrightarrow{a}+\overrightarrow{b}$ are collinear.

then $\overrightarrow{l}=\lambda \overrightarrow{m}$

$(2x+1)\overrightarrow{a}-\overrightarrow{b}=\lambda \left[\right(x-2)\overrightarrow{a}+\overrightarrow{b}]$

$(2x+1)\overrightarrow{a}-\overrightarrow{b}=\lambda (x-2)\overrightarrow{a}+\lambda \overrightarrow{b}$

on comparing we get

$[\lambda =-1]$

and $(2x+1)=\lambda (x-2)$

$2x+1=2-x$

$3x=1$

$[x=\frac{1}{3}]$

For $x=\frac{1}{3}\overrightarrow{l}$ & $\overrightarrow{m}$ are collinear.