Class CBSE Class 12 Mathematics Vector Algebra Q #1852
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$
(A) $6\sqrt{3}$
(B) $8\sqrt{3}$
(C) $12\sqrt{3}$
(D) $3\sqrt{12}$
Prev Finish
Correct Answer: C

AI Tutor Explanation

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Detailed Solution

Step 1: Identify the given values and formulas

We are given the magnitudes of two vectors and their cross product: $$|\vec{a}| = 8, |\vec{b}| = 3, |\vec{a} \times \vec{b}| = 12$$ We know the definitions: $$|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta = 12$$ $$|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos \theta$$

Step 2: Calculate sin θ

Substitute the given values into the cross product formula: $$8 \times 3 \times \sin \theta = 12$$ $$24 \sin \theta = 12$$ $$\sin \theta = \frac{12}{24} = \frac{1}{2}$$

Step 3: Determine cos θ

Using the identity $\sin^2 \theta + \cos^2 \theta = 1$: $$\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4}$$ $$\cos \theta = \frac{\sqrt{3}}{2}$$

Step 4: Calculate the dot product

Now substitute the values into the dot product formula: $$|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos \theta$$ $$|\vec{a} \cdot \vec{b}| = 8 \times 3 \times \frac{\sqrt{3}}{2}$$ $$|\vec{a} \cdot \vec{b}| = 24 \times \frac{\sqrt{3}}{2} = 12\sqrt{3}$$

Final Answer: 12\sqrt{3}

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must utilize trigonometric identities in conjunction with vector product definitions to derive the unknown value.
Knowledge Dimension: PROCEDURAL
Justification: The problem requires a sequential execution of mathematical steps (formula application, algebraic manipulation, and trigonometric identity usage).
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the conceptual understanding of the relationship between the dot and cross products of vectors, which is a core competency in the Vector Algebra unit.
||KEY:C

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