Question

For two vectors \( \vec{a} \) and \( \vec{b} \) Assertion (A) : \( |\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a|^2 |\vec{b}|^2 \)} Reason (R) : \( |\vec{a} \times \vec{b}| = (\vec{a} \cdot \vec{b}) \tan \theta, \theta \neq \frac{\pi{2} \)}

• 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:

Identity in (A) is known as Lagrange's Identity.
Reason (R) is simply a different way to express the relationship between sine and cosine components.

Answer 1

The Correct Option is A

Concept:
• Dot product: \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \).
• Cross product magnitude: \( |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \).
• Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).

Step 1: Evaluate Assertion (A)
Let \( |\vec{a}| = a \) and \( |\vec{b}| = b \).
LHS \( = (ab \sin \theta)^2 + (ab \cos \theta)^2 \)
LHS \( = a^2b^2 \sin^2 \theta + a^2b^2 \cos^2 \theta \)
LHS \( = a^2b^2 (\sin^2 \theta + \cos^2 \theta) = a^2b^2 \cdot 1 = a^2b^2 \)
RHS \( = a^2b^2 \).
Since LHS \( = \) RHS, Assertion (A) is true. (This is Lagrange's Identity).

Step 2: Evaluate Reason (R)
Starting from the RHS of Reason (R):
\( (\vec{a} \cdot \vec{b}) \tan \theta = (ab \cos \theta) \left( \frac{\sin \theta}{\cos \theta} \right) \)
\( = ab \sin \theta = |\vec{a} \times \vec{b}| \).
Reason (R) is true for \( \theta \neq \pi/2 \) (where tan is defined).

Step 3: Determine if (R) explains (A)
If we take Reason (R) and square it:
\( |\vec{a} \times \vec{b}|^2 = (\vec{a} \cdot \vec{b})^2 \tan^2 \theta \)
Adding \( (\vec{a} \cdot \vec{b})^2 \) to both sides:
\( |\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = (\vec{a} \cdot \vec{b})^2 (\tan^2 \theta + 1) \)
\( = (\vec{a} \cdot \vec{b})^2 \sec^2 \theta = (ab \cos \theta)^2 \cdot \frac{1}{\cos^2 \theta} = a^2b^2 \).
Since Reason (R) provides a direct relationship that leads to the truth of Assertion (A), it is the correct explanation.