Question

If three points A, B and $C$ have position vectors $(1,x,3)$, $(3,4,7)$ and $(y,-2,-5)$ respectively and, if they are collinear, then $(x,-y)$ is equal to

• A. (2,-3)
• B. (-2,3)
• C. (2,3)
• D. (-2,-3)

Answer 1

Answer: (C)

Given that $\overrightarrow{OA}=\hat{i}+\hat{x}+3\hat{k}$
$\overrightarrow{OB}=3\hat{i}+4\hat{j}+7\hat{k}$ and $\overrightarrow{OC}=y\hat{i}-2\hat{j}-5\hat{k}$
Since, $A,B,C$ are collinear, Then, $\overrightarrow{AB}=\lambda \overrightarrow{BC}$
$\Rightarrow 2\hat{i}+(4-x)\hat{j}+4\hat{k}=\lambda [(y-3)\hat{i}-6\hat{j}-12\hat{k}]$
On comparing the coefficients of $\hat{i},\hat{j}$ and $\hat{k}$,we get
$2=(y-3)\lambda \dots (i)$
$4-x=-6\lambda \dots (ii)$
and $4=-12\lambda$
$\Rightarrow \lambda =-\frac{1}{3}\dots (iii)$
On putting the value of $\lambda$ in eqs. (i) and (ii),we get
$y=-3$ and $x=2$