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): $|\vec a\times \vec b|=|\vec a||\vec b|\sin\theta$, where $\theta$ is the angle between them.

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

Hint:

Remember the important vector identity \(|\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2=|\vec a|^2|\vec b|^2\).

Answer 1

The Correct Option is A

To assess the given assertion and reason, we need to explore both statements and ascertain the relationship between them.

1. \(|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2\) is the assertion (A). This is a known mathematical identity involving vectors, often called the vector identity, and is indeed true.
2. The reason (R) states \(|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta\), where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\). This is a standard result for the magnitude of a cross product and is also true.

Let's verify the assertion using the given reason:

• We know that \(|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta\).
• Therefore, \(|\vec{a} \times \vec{b}|^2 = (|\vec{a}||\vec{b}|\sin\theta)^2 = |\vec{a}|^2 |\vec{b}|^2 \sin^2\theta\).
• Also, \((\vec{a} \cdot \vec{b}) = |\vec{a}||\vec{b}|\cos\theta\), hence \((\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2 \cos^2\theta\).
• The left-hand side of the assertion becomes:
  • \(|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2 \sin^2\theta + |\vec{a}|^2 |\vec{b}|^2 \cos^2\theta\)
• Factoring out \(|\vec{a}|^2 |\vec{b}|^2\) gives:
  • \(|\vec{a}|^2 |\vec{b}|^2 (\sin^2\theta + \cos^2\theta)\)
• Using the identity \(\sin^2\theta + \cos^2\theta = 1\), this simplifies to:
  • \(|\vec{a}|^2 |\vec{b}|^2\), confirming the assertion.

Thus, both the assertion (A) and the reason (R) are true. The reason is indeed the correct explanation for the assertion because it provides the necessary relationship through the known vector product properties.

Conclusion: Both (A) and (R) are true, and (R) is the correct explanation of (A).