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Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product5 Marks
Question
If either $\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0},$ then $\vec{\text{a}}.\vec{\text{b}}=0.$ But the converse need not be true. Justify your answer with an example.
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Answer

Let us assume that either $|\vec{\text{a}}|=0$ or $\big|\vec{\text{b}}\big|=0$Then, $\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$ ($\theta$ is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$)
Now, let us assume that $\vec{\text{a}}.\vec{\text{b}}=0$
$\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$
But here we cannot say that either $|\vec{\text{a}}|=0$ or $\big|\vec{\text{b}}\big|=0$. (Because even $\cos\theta$ can be zero)
For example, let
$\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{b}}=-3\hat{\text{i}}+2\hat{\text{k}}$
Here, $|\vec{\text{a}}|=\sqrt{4+1+9}=\sqrt{14}\neq0$
$\big|\vec{\text{b}}\big|=\sqrt{9+4}=\sqrt{13}\neq0$
But $\vec{\text{a}}.\vec{\text{b}}=\big(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}\big).\big(-3\hat{\text{i}}+2\hat{\text{k}}\big)=-6+0+6=0$

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