Question

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

Answer 1

Vectors $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ will be collinear, if

$\overrightarrow{\alpha }=m\overrightarrow{\beta }$ for some scalar m.

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

$\left[\right(2x+1)-m(x-2\left)\right]-(m+1)\overrightarrow{b}=\overrightarrow{0}$

$(2x+1)-m(x-2)=0$ and $-(m+1)=0$

$m=-1$ and $x=\frac{1}{3}$.