Question

The line whose vector equations are \(\vec{r}_1 = 2\hat{i} - 3\hat{j} + 7\hat{k} + \lambda(2\hat{i} + p\hat{j} + 5\hat{k})\) and \(\vec{r}_2 = \hat{i} + 2\hat{j} + 3\hat{k} + \mu(3\hat{i} - p\hat{j} + p\hat{k})\) are perpendicular for all values of \(\lambda\) and \(\mu\). The value of \(p\) is:

• A. \( -1 \)
• B. \( 2 \)
• C. \( 5 \)
• D. \( 6 \)

Hint:

For two lines to be perpendicular, their direction vectors must satisfy a dot product of zero.