Question

If \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \) and \( \vec{b} = p\hat{i} + 2\hat{j} + 2\hat{k} \) are perpendicular to each other, find the value of \(p\).

• A. \(4\)
• B. \(-4\)
• C. \(2\)
• D. \(-2\)

Hint:

Perpendicular vectors always satisfy: \[ \vec{a} \cdot \vec{b} = 0 \]

Answer 1

The Correct Option is B

Concept: Two vectors are perpendicular if their dot product is zero: \[ \vec{a} \cdot \vec{b} = 0 \]

Step 1: Applying the perpendicularity condition.
\[ (2\hat{i}+\hat{j}+3\hat{k}) \cdot (p\hat{i}+2\hat{j}+2\hat{k}) = 0 \]

Step 2: Expanding the dot product.
\[ 2p + 2 + 6 = 0 \]

Step 3: Solving for \(p\).
\[ 2p + 8 = 0 \] \[ 2p = -8 \] \[ p = -4 \]