Mathematics

Problem

The value of

a

, for which the points,

A

B

C

with position vectors

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 angled triangle with

C = \frac{\pi}{2}

are [AIEEE 2006] a. -2 and -1 b. -2 and 1 c. 2 and -1 d. 2 and 1

Intuitive explanation

Three points form a right angle at $C$ when the two sides meeting at $C$ are perpendicular—meaning their direction vectors have a dot product of zero. Find the vectors $\overrightarrow{CA}$ and $\overrightarrow{CB}$ in terms of $a$ , then set $\overrightarrow{CA} \cdot \overrightarrow{CB} = 0$ to solve for $a$ . This orthogonality condition is your key tool.

Intuitive explanation

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