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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
Find a vector whose length is 3 and which is perpendicular to the vector $\vec{\text{a}}=3\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$ and $\vec{\text{b}}=6\hat{\text{i}}+5\hat{\text{j}}-2\hat{\text{k}}.$
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Answer

vector perpendicular to $\vec{\text{a}}$ and $\vec{\text{b}}$ 
with magnitude $1=\frac{\vec{\text{a}}\times\vec{\text{b}}}{\big|\vec{\text{a}}\times\vec{\text{b}}\big|}$
$=\frac{1}{49}\big(7\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)\big)$
$=\frac{1}{7}\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)$
vector of magnitude 49, which is perpendicular to $\vec{\text{a}}$ and $\vec{\text{b}}$
$=49\Bigg(\frac{\vec{\text{a}}\times\vec{\text{b}}}{\big|\vec{\text{a}}\times\vec{\text{b}}\big|}\Bigg)$
$=49\big[\frac{1}{7}\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)\big]$
$=42\hat{\text{i}}+14\hat{\text{j}}-21\hat{\text{k}}$
The required vector $=42\hat{\text{i}}+14\hat{\text{j}}-21\hat{\text{k}}$

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