If the vectors $2 \hat{i}-\hat{j}-\hat{k} ; \hat{i}+2 \hat{j}-3 \hat{k}$ and $3 \hat{i}+\lambda \hat{j}+5 \hat{k}$ are coplanar, then the value of $\lambda$ is

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Aicharya · Accepted Answer

For given coplanar vectors, we write $\begin{aligned} & \left|\begin{array}{ccc} 2 & -1 & -1 \ 1 & 2 & -3 \ 3 & \lambda & 5 \end{array}ight|=\Rightarrow 2(10+3 \lambda)+(14)-(\lambda-6)=0 \ & \Rightarrow 5 \lambda+40=0 \Rightarrow \lambda=-8 \end{aligned}$

If the vectors $2 \hat{i}-\hat{j}-\hat{k} ; \hat{i}+2 \hat{j}-3 \hat{k}$ and $3 \hat{i}+\lambda \hat{j}+5…

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