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Question: Answered & Verified by Expert
If $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0}$ then which one of the following is
correct ?
MathematicsVector AlgebraNDANDA 2012 (Phase 2)
Options:
  • A $\overrightarrow{\mathrm{a}}$ is parallel to $\overrightarrow{\mathrm{b}}$
  • B $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}}$
  • C $\overrightarrow{\mathrm{a}}=\overrightarrow{0}$ or $\overrightarrow{\mathrm{b}}=\overrightarrow{0}$
  • D None of the above
Solution:
2434 Upvotes Verified Answer
The correct answer is: $\overrightarrow{\mathrm{a}}=\overrightarrow{0}$ or $\overrightarrow{\mathrm{b}}=\overrightarrow{0}$
$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0 \Rightarrow \overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}}$
$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=0 \Rightarrow \overrightarrow{\mathrm{a}} \| \overrightarrow{\mathrm{b}}$
(ii) it is clear that $\overrightarrow{\mathrm{a}}=0$ or $\overrightarrow{\mathrm{b}}=0$ From (i) and

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