NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

If $\mathbf{a}$ and $\mathbf{b}$ are vectors such that $|\mathbf{a}| = 13, |\mathbf{b}| = 5$ and $\mathbf{a} \cdot \mathbf{b} = 60$ , then the value of $|\mathbf{a} \times \mathbf{b}|$ is:

Choose the correct answer:

Explanation

1. Identify Given Information:

$|\mathbf{a}| = 13$
$|\mathbf{b}| = 5$
$\mathbf{a} \cdot \mathbf{b} = 60$

2. Apply Lagrange's Identity:

Lagrange's identity establishes the relationship between the dot product and the magnitude of the cross product:

|\mathbf{a} \times \mathbf{b}|^2 + (\mathbf{a} \cdot \mathbf{b})^2 = |\mathbf{a}|^2 |\mathbf{b}|^2

3. Substitute the values into the identity:

|\mathbf{a} \times \mathbf{b}|^2 + (60)^2 = (13)^2 (5)^2

|\mathbf{a} \times \mathbf{b}|^2 + 3600 = (169)(25)

|\mathbf{a} \times \mathbf{b}|^2 + 3600 = 4225

4. Solve for $|\mathbf{a} \times \mathbf{b}|$ :

|\mathbf{a} \times \mathbf{b}|^2 = 4225 - 3600

|\mathbf{a} \times \mathbf{b}|^2 = 625

|\mathbf{a} \times \mathbf{b}| = \sqrt{625}

|\mathbf{a} \times \mathbf{b}| = 25

Conclusion:

The value of $|\mathbf{a} \times \mathbf{b}|$ is 25. The correct option is (d).