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

Step 1: Recall vector product identities.
For two vectors \(\vec a\) and \(\vec b\) making an angle \(\theta\), the magnitudes of dot product and cross product are given by \[ \vec a\cdot\vec b=|\vec a||\vec b|\cos\theta \] and \[ |\vec a\times\vec b|=|\vec a||\vec b|\sin\theta \] Step 2: Square the expressions.
Squaring both expressions we get \[ (\vec a\cdot\vec b)^2=|\vec a|^2|\vec b|^2\cos^2\theta \] and \[ |\vec a\times\vec b|^2=|\vec a|^2|\vec b|^2\sin^2\theta \] Step 3: Add the two results.
Adding them gives \[ |\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2 \] \[ =|\vec a|^2|\vec b|^2(\sin^2\theta+\cos^2\theta) \] Step 4: Use trigonometric identity.
We know that \[ \sin^2\theta+\cos^2\theta=1 \] Therefore \[ |\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2 = |\vec a|^2|\vec b|^2 \] Thus the assertion is true.
Step 5: Analyze the reason.
The reason states the formula \[ |\vec a\times\vec b|=|\vec a||\vec b|\sin\theta \] This is the fundamental definition of the magnitude of the cross product and directly leads to the identity used in the assertion.
Therefore the reason correctly explains the assertion.
Step 6: Conclusion.
Both statements are correct and the reason explains the assertion.
Final Answer: $\boxed{\text{(A) Both (A) and (R) are true and (R) explains (A)}}$