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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
If A, B, C, D are the points with position vectors $\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}},\ 2\hat{\text{i}}-3\hat{\text{k}},$ respectively, find the projection of $\overrightarrow{\text{AB}}$ along $\overrightarrow{\text{CD}}.$
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Answer

We have $\overrightarrow{\text{AB}}=\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ \overrightarrow{\text{OB}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}},$
$\overrightarrow{\text{OC}}=\ 2\hat{\text{i}}-3\hat{\text{k}}$ and $\overrightarrow{\text{OD}}=3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$
$\therefore\overrightarrow{\text{AB}}=\overrightarrow{\text{OB}}-\overrightarrow{\text{OA}}$
$=(2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}})-(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})$
$=\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}$
and $\overrightarrow{\text{CD}}=\overrightarrow{\text{OD}}-\overrightarrow{\text{OC}}$
$=(3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}})-(2\hat{\text{i}}-3\hat{\text{k}})$
$=\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}$
Now the projection of $\overrightarrow{\text{AB}}$ along $\overrightarrow{\text{CD}}$
$=\overrightarrow{\text{AB}}.\frac{\overrightarrow{\text{CD}}}{|\overrightarrow{\text{CD}|}}$
$=\frac{(\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}})\cdot(\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}})}{\sqrt{1^1+2^2+4^2}}$
$=\frac{1+4+16}{\sqrt{21}}=\frac{21}{\sqrt{21}}$
$=\sqrt{21}\text{ units}$

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