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 $AB$ and $CD$ is $\frac{\pi }{2}$.
Statement II $AB$ and $CD$ 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 $AB$ and $CD$, then $\theta$ is also the angle between $AB$ and $CD$.
Now, $AB=$ 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, $|AB|=\sqrt{1^2+4^2+(-1{)}^2}=3\sqrt{2}$
Similarly, $(CD)=-2\hat{i}-8\hat{j}+2\hat{k}$ and $|CD|=6\sqrt{2}$
$\therefore \cos \theta =\frac{AB\cdot CD}{|AB||CD|}$
$=\frac{1(-2)+4(-8)+(-1)(2)}{(3\sqrt{2})(6\sqrt{2})}$
$=\frac{-36}{36}=-1$
Since, $0\le \theta \le \pi$, it follows that $\theta =\pi$.
This shows that $AB$ and $CD$ are collinear.
Alternatively, $AB=-\frac{1}{2}CD$ which implies that $AB$ and $CD$ are collinear vectors.
So, only statement II is true.