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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA1 Mark
Question
Fill in the blanks.
If $\vec{\text{a}}$ is any non-zero vector, then $(\vec{\text{a}}\cdot\vec{\text{i}})\vec{\text{i}}+(\vec{\text{a}}\cdot\vec{\text{j}})\vec{\text{j}}+(\vec{\text{a}}\cdot\vec{\text{k}})\vec{\text{k}}$ equal ________.
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Answer

If $\vec{\text{a}}$ is any non-zero vector, then $(\vec{\text{a}}\cdot\vec{\text{i}})\vec{\text{i}}+(\vec{\text{a}}\cdot\vec{\text{j}})\vec{\text{j}}+(\vec{\text{a}}\cdot\vec{\text{k}})\vec{\text{k}}$ equal $\vec{\text{a}}={\text{a}}_1\hat{{\text{i}}}+{\text{a}}_2\hat{{\text{j}}}+{\text{a}}_3\hat{{\text{k}}}.$Solution:
$\vec{\text{a}}={\text{a}}_1\hat{{\text{i}}}+{\text{a}}_2\hat{{\text{j}}}+{\text{a}}_3\hat{{\text{k}}}$
Taking dot producrt with $\hat{\text{i}},$ we get ${\text{a}}_1\cdot\hat{{\text{i}}}={\text{a}}_1\hat{{\text{i}}}+0+0$
$\Rightarrow{\text{a}}_1=\vec{\text{a}}\cdot\hat{{\text{i}}}$
Similarly taking dot product with $\hat{{\text{j}}}$ and $\hat{{\text{k}}},$ we get
${\text{a}}_1=\vec{\text{a}}\cdot\hat{{\text{j}}}=\text{a}_2$ and $\vec{\text{a}}\cdot\hat{{\text{k}}}={\text{a}}_3$
$\therefore\vec{\text{a}}={\text{a}}_1\hat{{\text{i}}}+{\text{a}}_2\hat{{\text{j}}}+{\text{a}}_3\hat{{\text{k}}}$ $=(\vec{\text{a}}\cdot\vec{\text{i}})\vec{\text{i}}+(\vec{\text{a}}\cdot\vec{\text{j}})\vec{\text{j}}+(\vec{\text{a}}\cdot\vec{\text{k}})\vec{\text{k}}$

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