Question

If the position vectors of the points $A$, $B$, and $C$ be $i+j$ , $i-j$ and $ai+bj+ck$ respective;y , then the points $A$, $B$ and $C$ are collinear if:

• A. $a=b=c=1$
• B. $a=1$ , $b$ and $c$ are arbitrary scalars
• C. $a=b=c=0$
• D. $c=0$ , $a=1$ and $b$ is a arbitrary scalar.

Answer 1

Answer: (D)

$c=0$ , $a=1$ and $b$ is a arbitrary scalar.

Given points are $A(i+j)$ , $B(i-j)$ and $C(ai+bj+ck)$

These three points are collinear if drs of lines $AB$ and $CB$ are proportional

We have $AB=2j$ and $CB=(a-1)i+(b+1)j+ck$

Since their drs are proportional, we get $\frac{a-1}{0}=\frac{b+1}{2}=\frac{c}{0}$

$\Rightarrow c=0,a=1$ and $b$ can be a arbitrary scalar

Therefore option $D$ is correct