Class CBSE Class 12 Mathematics Vector Algebra Q #570
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
If vector \(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\) and vector \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\), then which of the following is correct ?
(A) \(\vec{a} \parallel \vec{b}\)
(B) \(\vec{a} \perp \vec{b}\)
(C) \(|\vec{b}| > |\vec{a}|\)
(D) \(|\vec{a}| = |\vec{b}|\)
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Correct Answer: B
Explanation
**Correct Option if MCQ:** B
**Reasoning:**
* Calculate the dot product: \(\vec{a} \cdot \vec{b} = (3)(1) + (2)(-1) + (-1)(1) = 3 - 2 - 1 = 0\).
* Since the dot product is zero, the vectors are perpendicular.
* \(\vec{a} \perp \vec{b}\)

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Step-by-Step Solution

  1. Check for Parallelism: Two vectors are parallel if one is a scalar multiple of the other. Here, \(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\). There is no scalar \(k\) such that \(\vec{a} = k\vec{b}\). So, \(\vec{a}\) is not parallel to \(\vec{b}\).

  2. Check for Perpendicularity: Two vectors are perpendicular if their dot product is zero. \(\vec{a} \cdot \vec{b} = (3)(1) + (2)(-1) + (-1)(1) = 3 - 2 - 1 = 0\). Therefore, \(\vec{a} \perp \vec{b}\).

  3. Calculate Magnitudes: \(|\vec{a}| = \sqrt{(3)^2 + (2)^2 + (-1)^2} = \sqrt{9 + 4 + 1} = \sqrt{14}\) \(|\vec{b}| = \sqrt{(1)^2 + (-1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}\)

  4. Compare Magnitudes: Since \(\sqrt{14} \neq \sqrt{3}\), \(|\vec{a}| \neq |\vec{b}|\). Also, since \(\sqrt{3} < \sqrt{14}\), \(|\vec{b}| < |\vec{a}|\).

  5. Conclusion: From the above steps, we find that \(\vec{a} \perp \vec{b}\) is the correct statement.

Correct Answer: \(\vec{a} \perp \vec{b}\)

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of vectors, dot product, and magnitude to determine the relationship between the given vectors.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to calculate the dot product and magnitudes of the vectors to determine their relationship.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of vector algebra concepts as covered in the textbook.

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