Question

If the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear, then the value of $x$, for which, the vectors $\overrightarrow{c}=(x-2)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{d}=(2x+1)\overrightarrow{a}-\overrightarrow{b}$ are collinear, is equal to

• A. $\frac{4}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. none of these

Answer 1

Answer: (C)

$\frac{1}{3}$

The vector $c$ is non-zero since the coefficient in $b$ is different from zero,

and so the vectors $c$ and $d$ are collinear if for some number $y$ we have

$d=yc$ that is

$(2x+1)a-b=y(x-2)a+yb$

or $(yx-2y-2x-1)a+(y+1)b=0.$

Since $a,b$ are non-collinear, we must have

$yx-2y-2x-1=0$ and $y+1=0.$

Solving these equations, we get $y=-1$ and $x=\frac{1}{3}$