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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
$\text{If} \ \vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\ \ \vec{b}=-$ $\hat{i}+2\hat{j}+\hat{k}\ \text{and}\ \vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c},$ then find the value of $\lambda.$
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Answer

$\text{Given:}\ \ \ \ \vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\ \vec{b}=-$ $\hat{i}+2\hat{j}+\hat{k}\ \text{and}\ \vec{c}=3\hat{i}+\hat{j}$$\text{Now}\ \ \ \ \vec{a}+\lambda\vec{b}=2\hat{i}+2\hat{j}+3\hat{k}+\lambda(-\hat{i}+2\hat{j}+\hat{k})$ $=2\hat{i}+2\hat{j}+3\hat{k}-\lambda\hat{i}+2\lambda\hat{j}+\lambda\hat{k}$
$\Rightarrow\ \ \vec{a}+\lambda\vec{b}=(2-\lambda)\hat{i}+(2+2\lambda)\hat{j}+(3+\lambda)\hat{k}$
$\text{Again},\ \vec{c}=3\hat{i}+\hat{j}=3\hat{i}+\hat{j}+0\hat{k}$
$\text{Since},\ (\vec{a}+\lambda\vec{b})$ is perpendicular to $\vec{c},$ therefore, $\ \ (\vec{a}+\lambda\vec{b}).\vec{c}=0$
$\Rightarrow\ (2-\lambda)3+(2+2\lambda)1+(3+\lambda)0=0$
$\Rightarrow\ 6-3\lambda+2+2\lambda=0$ $\Rightarrow\ -\lambda+8=0$
$\Rightarrow\ -\lambda=-8\ \ \ \Rightarrow\ \lambda=8$

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