Discussion
Let \(\vec{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k})\) = \((2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{b}\). If \(|\vec{a} + \vec{b}| = \sqrt{29}\) ,then \(\vec{a} + \vec{b}\) =
Answer: C
\(\vec{a} + \vec{b} = \pm(2\hat{i} + 3\hat{j} + 4\hat{k})\).
Let \(\vec{c} = 2\hat{i} + 3\hat{j} + 4\hat{k}\). Since \(\vec{c} \times \vec{b} = -\vec{b} \times \vec{c}\), the condition becomes
\[ \begin{aligned} \vec{a} \times \vec{c} + \vec{b} \times \vec{c} &= (\vec{a} + \vec{b}) \times \vec{c} \\ &= \vec{0} \end{aligned} \]
so \(\vec{a} + \vec{b}\) is parallel to \(\vec{c}\). Its magnitude \(\sqrt{29}\) equals \(|\vec{c}| = \sqrt{4 + 9 + 16}\), so \(\vec{a} + \vec{b} = \pm\vec{c}\).
The vectors with a changed sign on one component are not parallel to \(\vec{c}\), and \(\pm\sqrt{29}\,\vec{c}\) has magnitude 29.
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