question_answer

The value of a for which the points A, B, C with position vector \[2\hat{i}-\hat{j}+\hat{k},\] \[\hat{i}-3\hat{j}-5\hat{k},\] and \[a\hat{i}-3\hat{j}+\hat{k}\] respectively are the vertices of a right angle triangle with \[C=\frac{\pi }{2}\] are

A) 2 and 1             

B) \[-\,2\,\,\text{and}\,-\,1\]

C) \[-\,2\,\,\text{and}\,\,1\]              

D) \[2\,\,\text{and}\,\,-1\]

Correct Answer: A

Solution :

[a]

We have,

\[A=2\hat{i}-\hat{j}+\hat{k},\]\[B=\hat{i}-3\hat{j}-5\hat{k},\]\[C=a\hat{i}-3\hat{j}+\hat{k}\]

\[CA=(2-a)\hat{i}+2\hat{j}\]

\[CB=(1-a)\hat{i}-6\hat{k}\]

\[CA\cdot CB=0\]

\[\therefore \]\[(2-a)(1-a)=0\]

\[a=2,1\]