Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product3 Marks
Question
What inference can you draw if $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$ and $\vec{\text{a}}.\vec{\text{b}}=0.$
✓
Answer
Given: $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\vec{0}$ $\Rightarrow\vec{\text{a}}=0$ $\vec{\text{b}}=0$ $\therefore\vec{\text{a}}||\vec{\text{b}}$ Also, $\vec{\text{a}}.\vec{\text{b}}=0$ $\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$ $\Rightarrow\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0}$ or, $\vec{\text{a}}\perp\vec{\text{b}}$ But $\vec{\text{a}}$ cannot be both perpendicular as well as parallel to $\vec{\text{b}}.$ $\therefore|\vec{\text{a}}|=0$ $\big|\vec{\text{b}}\big|=0$
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