Question:
The value of \( p \) for which vectors \( \hat{i} + 2\hat{j} + 3\hat{k} \) and \( 2\hat{i} - p\hat{j} + \hat{k} \) are perpendicular to each other is
The value of \( p \) for which vectors \( \hat{i} + 2\hat{j} + 3\hat{k} \) and \( 2\hat{i} - p\hat{j} + \hat{k} \) are perpendicular to each other is
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Perpendicularity check: \( \vec{a} \cdot \vec{b} = 0 \).
Parallelism check: \( \vec{a} \times \vec{b} = 0 \) or components are proportional.
Parallelism check: \( \vec{a} \times \vec{b} = 0 \) or components are proportional.
Updated On: Sep 10, 2026
- \( 0 \)
- \( 1 \)
- \( \frac{5}{2} \)
- \( -\frac{5}{2} \)
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The Correct Option is C
Solution and Explanation
Concept:
• Two non-zero vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular if and only if their dot product (scalar product) is zero.
• If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
Step 1: Set up the dot product equation
Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - p\hat{j} + \hat{k} \). Since the vectors are perpendicular: \[ \vec{a} \cdot \vec{b} = 0 \]
Step 2: Calculate the dot product
Multiply the corresponding components of the vectors: \[ (1)(2) + (2)(-p) + (3)(1) = 0 \] \[ 2 - 2p + 3 = 0 \]
Step 3: Solve for \( p \)
Combine the numerical terms: \[ 5 - 2p = 0 \] Isolate \( p \): \[ 2p = 5 \] \[ p = \frac{5}{2} \]
• Two non-zero vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular if and only if their dot product (scalar product) is zero.
• If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
Step 1: Set up the dot product equation
Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - p\hat{j} + \hat{k} \). Since the vectors are perpendicular: \[ \vec{a} \cdot \vec{b} = 0 \]
Step 2: Calculate the dot product
Multiply the corresponding components of the vectors: \[ (1)(2) + (2)(-p) + (3)(1) = 0 \] \[ 2 - 2p + 3 = 0 \]
Step 3: Solve for \( p \)
Combine the numerical terms: \[ 5 - 2p = 0 \] Isolate \( p \): \[ 2p = 5 \] \[ p = \frac{5}{2} \]
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