Question

Let $A,B,C$ be distinct point with position vectors $\hat{i}+\hat{j}$, $\hat{i}-\hat{j}$, $p\hat{i}-q\hat{j}+r\hat{k}$ respectively. Points $A,B,C$ are collinear, then which of the following can be correct:

• A. $p=q=r=1$
• B. $p=q=r=0$
• C. $p=q=2,r=0$
• D. $p=1,q=2,r=0$

Answer 1

The correct option is D $p=1,q=2,r=0$

The three points A, B and C are collinear if they satisfy one of the conditions

$1)AB+BC=AC,2).AC+BC=AB$ and $3).AB+AC=BC$

considering the first condition, we have

$AB=2$, $AC=\sqrt{(1-p{)}^2+(1+q{)}^2+r^2}$ and $BC=\sqrt{(1-p{)}^2+(-1+q{)}^2+r^2}$

To satisfy the condition $AB+BC=AC$, $p,q$ and $r$ take the values $1,2$ and $0$ respectively.

Option D gives the correct answer.