Question

The lines whose vector equations are
$r = 2\hat{i} - 3 \hat{j} + 7 \hat{k} + \lambda (2 \hat{i} + p\hat{j} + 5 \hat{k})$
and $r = \hat{i} - 2 \hat{j} + 3 \hat{k} + \mu (3 \hat{i} + p \hat{j} + p \hat{k})$
are perpendicular for all values of $\lambda$ and $\mu$ if p =

• A. 1
• B. -1
• C. -6
• D. 6

Answer 1

Answer: (D)

$2 \hat{i} - p\hat{j}+ 5 \hat{k} $ and $3\hat{i} + p\hat{j} + p\hat{k}$ are perpendicular
$\Rightarrow \, 2 \times 3 + p (-p) + 5 (p) = 0$
$\Rightarrow \, p^2 - 5p - 6 = 0 \, \Rightarrow \, p = -1$ or $p = 6$
Hence for p = 6, the lines are perpendicular.