Question

If the position vectors of the points A, B, C be and $a \mathbf { i } + b \mathbf { j } + c \mathbf { k }$ respectively, then the points A, B, C are collinear if

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

Answer 1

Answer: (D)

$c = 0 , a = 1$ and b is arbitrary scalars

Answer 2

Here $\overrightarrow { A B } = - 2 \mathbf { j }$ $\overrightarrow { B C } = ( a - 1 ) \mathbf { i } + ( b + 1 ) \mathbf { j } + c \mathbf { k }$

The points are collinear, then $\overrightarrow { A B } = k ( \overrightarrow { B C } )$

$- 2 \mathbf { j } = k \{ ( a - 1 ) \mathbf { i } + ( b + 1 ) \mathbf { j } + c \mathbf { k } \}$

On comparing, $k ( a - 1 ) = 0$, $k ( b + 1 ) = - 2$ $k c = 0$.

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