Class CBSE Class 12 Mathematics Vector Algebra Q #557
KNOWLEDGE BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:
(A) \(2\hat{j}\)
(B) \(\hat{j}\)
(C) \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)
(D) \(\frac{\hat{i}+\hat{k}}{\sqrt{2}}\)
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AI Tutor Explanation

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Step-by-Step Solution

  1. Let \(\vec{a} = \hat{i} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{k}\).

  2. Find a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) by calculating their cross product: \(\vec{a} \times \vec{b}\).

    \(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 1 \\ 1 & 0 & -1 \end{vmatrix} = \hat{i}(0 - 0) - \hat{j}(-1 - 1) + \hat{k}(0 - 0) = 2\hat{j}\)

  3. Find the unit vector in the direction of \(2\hat{j}\).

    The magnitude of \(2\hat{j}\) is \(|2\hat{j}| = 2\).

    The unit vector is \(\frac{2\hat{j}}{2} = \hat{j}\).

Correct Answer: \(\hat{j}\)

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 concept of finding a vector perpendicular to two given vectors using the cross product and then normalizing it to find the unit vector.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure: calculating the cross product of two vectors and then normalizing the resulting vector.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of vector algebra concepts and their ability to apply standard procedures to solve a problem.

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