CBSE Class 12 Mathematics Vector Algebra Q #1762
COMPETENCY BASED
APPLY
2 Marks 2026 AISSCE(Board Exam) VSA
Let two rods placed on the ground be represented by vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.
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Detailed Solution

Step 1: Identify the direction of the flag-post

Since the flag-post is perpendicular to both rods represented by vectors a = 4i - j + 3k and b = -2i + j - 2k, its direction must be parallel to the cross product n = a × b.

Step 2: Calculate the cross product

The cross product is calculated using the determinant: $$n = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & -1 & 3 \\ -2 & 1 & -2 \end{vmatrix}$$ $$n = \hat{i}((-1)(-2) - (3)(1)) - \hat{j}((4)(-2) - (3)(-2)) + \hat{k}((4)(1) - (-1)(-2))$$ $$n = \hat{i}(2 - 3) - \hat{j}(-8 + 6) + \hat{k}(4 - 2) = -\hat{i} + 2\hat{j} + 2\hat{k}$$

Step 3: Find the unit vector and scale to height

The magnitude of n is |n| = sqrt((-1)² + 2² + 2²) = sqrt(1 + 4 + 4) = 3. The unit vector is u = (-1/3)i + (2/3)j + (2/3)k. For a height of 5 m, the vector is 5u.

Final Answer: ± 5/3 (-i + 2j + 2k)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must utilize the geometric property of the cross product to find a normal vector and then scale it to a specific magnitude.
Knowledge Dimension: PROCEDURAL
Justification: The problem requires a multi-step algorithmic process involving vector cross-product calculation, normalization, and scalar multiplication.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the application of Vector Algebra (Chapter 10) in a real-world spatial context.

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