Given below are two statements:
Statement I : The angle between the vectors \( 2 \hat{\imath}+3 \hat{\jmath}+ \) \( \hat{k} \) and \( 2 \hat{\imath}-\hat{\jmath}-\hat{k} \) is \( \pi / 2 \) Statement II :The vector \( \vec{\alpha} \times(\vec{b} \times \vec{c}) \) is coplanar with \( \vec{a} \) and \( \vec{b} \) In the light of the above statement, choose the correct the answer from the options given below:Step-by-step Solution:
Statement I:
The angle between vectors \( 2\hat{\imath} + 3\hat{\jmath} + \hat{k} \) and \( 2\hat{\imath} - \hat{\jmath} - \hat{k} \) is \( \pi/2 \).
- Verification: Dot product \( (2)(2) + (3)(-1) + (1)(-1) = 0 \).
- Conclusion: True (vectors are perpendicular).
Statement II:
The vector \( \vec{\alpha} \times (\vec{b} \times \vec{c}) \) is coplanar with \( \vec{a} \) and \( \vec{b} \).
- Verification: By vector identity, the result lies in the plane of \( \vec{b} \) and \( \vec{c} \), not necessarily \( \vec{a} \) and \( \vec{b} \).
- Conclusion: False (only true if \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \)).