Question

The two vectors $a$ and $b$ are non-collinear then at what value of $x$ the vectors $c=(2x-3)a+b$ and $d=(2x+5)a-b$ are collinear ?

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

Answer 1

Answer: (A)

$\frac{-1}{2}$

Given vectors $c$ and $d$ are collinear

So we have $c=\lambda d$ , for some $\lambda$

$\Rightarrow (2x-3)a+b=\lambda (2x+5)a-\lambda b$

$\Rightarrow \left(\right(2\lambda -2)x+5\lambda +3)a-(1+\lambda )b=0$

Given that $a$ and $b$ are not collinear , so we get

$\Rightarrow \left(\right(2\lambda -2)x+5\lambda +3)=0$ and $1+\lambda =0$

So we get $\lambda =-1$

If we substitute the value of $\lambda$ in other equation , we get $(-4)x-2=0$

$\Rightarrow x=-\frac{1}{2}$

Therefore the correct option is $A$