Question

Assertion (A): If $| \mathbf{a} \times \mathbf{b} |^2 + | \mathbf{a} \cdot \mathbf{b} |^2 = 256$ and $| \mathbf{b} | = 8$, then $| \mathbf{a} | = 2$.
Reason (R): $\sin^2 \theta + \cos^2 \theta = 1$ and $| \mathbf{a} \times \mathbf{b} | = | \mathbf{a} | | \mathbf{b} | \sin \theta$ and $ \mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | | \mathbf{b} | \cos \theta$.

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

For problems involving cross and dot products, remember the identity $\sin^2 \theta + \cos^2 \theta = 1$, which simplifies the equations involving both products.

Answer 1

The Correct Option is A

The equation provided is \[ | \mathbf{a} \times \mathbf{b} |^2 + | \mathbf{a} \cdot \mathbf{b} |^2 = 256. \] By utilizing the properties of cross and dot products, we can rewrite this as: \[ | \mathbf{a} \times \mathbf{b} |^2 = | \mathbf{a} |^2 | \mathbf{b} |^2 \sin^2 \theta \quad \text{and} \quad | \mathbf{a} \cdot \mathbf{b} |^2 = | \mathbf{a} |^2 | \mathbf{b} |^2 \cos^2 \theta. \] Substituting these into the original equation yields: \[ | \mathbf{a} |^2 | \mathbf{b} |^2 (\sin^2 \theta + \cos^2 \theta) = 256. \] Given that $\sin^2 \theta + \cos^2 \theta = 1$, the equation simplifies to: \[ | \mathbf{a} |^2 | \mathbf{b} |^2 = 256. \] With the information that $| \mathbf{b} | = 8$, we can substitute this value: \[ | \mathbf{a} |^2 (8)^2 = 256 \quad \Rightarrow \quad | \mathbf{a} |^2 \times 64 = 256 \quad \Rightarrow \quad | \mathbf{a} |^2 = 4 \quad \Rightarrow \quad | \mathbf{a} | = 2. \] Therefore, both Assertion (A) and Reason (R) are correct, with Reason providing a valid explanation for Assertion.