If $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \quad \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\bar{c}=3 \hat{i}+\hat{j}$ are the vectors such that $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$, then value of $\lambda$ is

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Aicharya · Accepted Answer

$\begin{aligned} & \overline{\mathrm{a}}+\lambda \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \ & \quad=(2-\lambda) \hat{\mathrm{i}}+(2+2 \lambda) \hat{\mathrm{j}}+(3+\lambda) \hat{\mathrm{k}} \ & (\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}}) \cdot \overline{\mathrm{c}}=0 \ & \Rightarrow(2-\lambda)(3)+(2+2 \lambda)(1)+(3+\lambda)(0)=0 \ & \Rightarrow 8-\lambda=0 \ & \Rightarrow \lambda=8\end{aligned}$

If $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \quad \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\bar{c}=3…

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