The value of p for which the vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ are perpendicular to each other is

Concept: Dot product = 0 is the necessary and sufficient condition for perpendicularity of two non-zero vectors.

For $a = i+2j+3k$ and $b = 2i-pj+k$, $a \cdot b = (1)(2) + (2)(-p) + (3)(1) = 2 - 2p + 3 = 5 - 2p$. Perpendicularity requires $a \cdot b = 0$, so $5 - 2p = 0$, giving $p = \frac{5}{2}$.

Why the distractors fail:

• (A) 0 would make a.b = 5, not 0, so the vectors would not be perpendicular; this likely comes from forgetting to solve the equation and picking a 'default' value.
• (B) 1 gives a.b = 5 - 2 = 3, not 0, an arithmetic slip in solving $5 - 2p = 0$.
• (D) -5/2 is the correct magnitude with the wrong sign, typically from an error transposing terms across the equation (writing 5 + 2p = 0 instead of $5 - 2p = 0$).

Exam tip: Carefully track the sign of each component when computing the dot product component-wise — a single sign slip changes the final answer completely.