Question

Vectors $\overline{a},\overline{b}$ are non-collinear find x.,so that the vectors $(x-1)\overline{a}+\overline{b}and(2+3x)\overline{a}-2\overline{b}$ are collinear.

Answer 1

$(x-1)\overline{a}+\overline{b}$ and $(2+3x)\overline{a}-2\overline{b}$ are collinear

$\Rightarrow k\left[\right(x-1)\overline{a}+\overline{b}]=(2+3x)\overline{a}-2\overline{b}$

$\Rightarrow \left[k\right(x-1\left)\right]\overline{a}+k\overline{b}=(2+3x)\overline{a}-2\overline{b}$

On comparing

$k=-2$ and $k(x-1)=(2+3x)$

$\Rightarrow -2(x-1)=2+3x$

$\Rightarrow -2x+2=2+3x$

$\Rightarrow 5x=0\Rightarrow x=0$

$\therefore (x-1)\overline{a}+\overline{b}$ and $(2+3x)\overline{a}-2\overline{b}$ are colliniear if $x=0$