Discussion
\(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) be non-zero vectors such that \(\vec{a} \times \vec{b} = \vec{c} \times \vec{d}\) and \(\vec{a} \times \vec{c} = \vec{b} \times \vec{d}\).Then
Answer: A
The conclusion is \(\vec{a} - \vec{d}\) is parallel to \(\vec{b} - \vec{c}\).
Two non-zero vectors are parallel when their cross product is zero. Expand \((\vec{a} - \vec{d}) \times (\vec{b} - \vec{c})\).
\[ \begin{aligned} &\vec{a} \times \vec{b} - \vec{a} \times \vec{c} - \vec{d} \times \vec{b} + \vec{d} \times \vec{c} \\ &= \vec{a} \times \vec{b} - \vec{a} \times \vec{c} + \vec{b} \times \vec{d} - \vec{c} \times \vec{d} \end{aligned} \]
The given conditions make the first and last terms cancel (\(\vec{a} \times \vec{b} = \vec{c} \times \vec{d}\)) and the middle two cancel (\(\vec{a} \times \vec{c} = \vec{b} \times \vec{d}\)), so the cross product is zero. The other pairings do not reduce to these two conditions.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development