Find the value of a ,b and c coplanar, where a = 2i -j + k, b = i… - Vidyadip

Question

Find the value of $\lambda$ $\overrightarrow{a} ,\overrightarrow{b}$ and $\overrightarrow{c}$ coplanar, where $\overrightarrow{a} = 2\hat{i} -\hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} + 2\hat{j} - 3\hat{k} $ and $\overrightarrow{c} = 3\hat{i}- \lambda \hat{j} + 5\hat{k}.$

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Answer

$\overrightarrow{a} = 2\hat{i} -\hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} + 2\hat{j} - 3\hat{k}, \overrightarrow {c} = 3\hat{i}- \lambda \hat{j} + 5\hat{k}.$ are coplanar if $\bigg[\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \bigg] = 0$
$ \begin{vmatrix} 2 & -1 & 1 \\ 1 & 2 & -3 \\ 3 & -\lambda & 5 \end{vmatrix} = 0 $
$\Rightarrow 2 (10 - 3 \lambda) + 1 (5 + 9) + ( -\lambda - 6) =0$
$\Rightarrow 7 \lambda = 28$
OR
$\lambda = 4$

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