Question

Assertion (A): $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.
Reason (R): $|\vec{a} \times \vec{b}| = (\vec{a} \cdot \vec{b})\tan\theta$, where $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ ($\theta \neq \frac{\pi}{2}$).

• 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:

Lagrange's Identity $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$ is incredibly useful for finding cross-product magnitudes when only the dot product and vector lengths are given!

Answer 1

The Correct Option is A

Concept: Let $\theta$ be the angle between two non-zero vectors $\vec{a}$ and $\vec{b}$. By definition:
• Dot product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$
• Cross product magnitude: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ Combining these products helps establish Lagrange's Identity.

Step 1: Evaluate the Assertion using product definitions.
Let us substitute the definitions of dot and cross products into the left-hand side of the assertion: \[ \text{LHS} = (|\vec{a}||\vec{b}|\sin\theta)^2 + (|\vec{a}||\vec{b}|\cos\theta)^2 \] Squaring the individual terms: \[ \text{LHS} = |\vec{a}|^2|\vec{b}|^2\sin^2\theta + |\vec{a}|^2|\vec{b}|^2\cos^2\theta \] Factoring out the common term $|\vec{a}|^2|\vec{b}|^2$: \[ \text{LHS} = |\vec{a}|^2|\vec{b}|^2 (\sin^2\theta + \cos^2\theta) \] Since $\sin^2\theta + \cos^2\theta = 1$: \[ \text{LHS} = |\vec{a}|^2|\vec{b}|^2 \cdot 1 = |\vec{a}|^2|\vec{b}|^2 = \text{RHS} \] Thus, the Assertion is true. This identity is known as Lagrange's Identity.

Step 2: Evaluate the Reason statement.
Let us look at the ratio of cross-product magnitude to dot product: \[ \frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}} = \frac{|\vec{a}||\vec{b}|\sin\theta}{|\vec{a}||\vec{b}|\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta \] Cross-multiplying gives: \[ | \vec{a} \times \vec{b} | = (\vec{a} \cdot \vec{b})\tan\theta \] This statement is true. Furthermore, since squaring this relationship directly leads to the identity verified in Step 1 ($\tan^2\theta = \sec^2\theta - 1$), the Reason serves as a valid explanation for the Assertion. Therefore, both statements are true, and the Reason is the correct explanation. This corresponds to option (A).