Question

It is given that position vectors of points $A, B, C$ and $D$ are $\hat{ i }+\hat{ j }+\hat{ k }, 2 \hat{ i }+5 \hat{ j }, 3 \hat{ i }+2 \hat{ j }-3 \hat{ k }$ and $\hat{ i }-6 \hat{ j }-\hat{ k }$ respectively.
Statement I The angle between $A B$ and $C D$ is $\frac{\pi}{2}$.
Statement II $A B$ and $C D$ are collinear vectors.

• A. Statement I is true; statement II is true; statement II is a correct explanation for statement I
• B. Statement I is true; statement II is true; statement II is not a correct explanation for statement I
• C. Statement I is true; statement II is false
• D. Statement I is false; statement II is true

Answer 1

Answer: (D)

If $\theta$ is the angle between $A B$ and $C D$, then $\theta$ is also the angle between $A B$ and $C D$.
Now, $A B =$ Position vector of $B$ - Position vector of $A$
$=(2 \hat{i}+5 \hat{j})-(\hat{i}+\hat{j}+\hat{k})=\hat{i}+4 \hat{j}-\hat{k}$
Therefore, $|A B|=\sqrt{1^2+4^2+(-1)^2}=3 \sqrt{2}$
Similarly, $(C D)=-2 \hat{i}-8 \hat{j}+2 \hat{k}$ and $|C D|=6 \sqrt{2}$
$\therefore \cos \theta =\frac{A B \cdot C D}{|A B||C D|} $
$ =\frac{1(-2)+4(-8)+(-1)(2)}{(3 \sqrt{2})(6 \sqrt{2})} $
$=\frac{-36}{36}=-1$
Since, $0 \leq \theta \leq \pi$, it follows that $\theta=\pi$.
This shows that $A B$ and $C D$ are collinear.
Alternatively, $A B=-\frac{1}{2} C D$ which implies that $A B$ and $C D$ are collinear vectors.
So, only statement II is true.