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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$
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Answer

Given:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$
$\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$
We know that vectors $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar iff $\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]=0.$
It is given that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar.
$\therefore\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]=0$
$\Rightarrow\begin{vmatrix}2&-1&1\\1&2&-3\\\lambda&\lambda&5 \end{vmatrix}=0$
$\Rightarrow2(10+3\lambda)+1(5+3\lambda)+1(\lambda-2\lambda)=0$
$\Rightarrow 8\lambda +25=0$
$=-\frac{25}{8}$

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