Given three vectors $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, a vector in the plane of $\mathbf{b}$ and $\mathbf{c}$ whose…

Question

Given three vectors $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, a vector in the plane of $\mathbf{b}$ and $\mathbf{c}$ whose projection on $\mathbf{a}$ is of magnitude $\sqrt{\frac{2}{3}}$ is

Aicharya · Accepted Answer

Given three vectors are $\begin{aligned} \mathbf{a} & =2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}} \ \text {and } \quad \mathbf{c} & =\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}} \end{aligned}$ Now, let a vector in plane of $\mathbf{b}$ and $\mathbf{c}$ is $\begin{aligned} \mathbf{d} & =\mathbf{b}+\lambda \mathbf{c} \ & =(1+\lambda) \hat{\mathbf{i}}+(2+\lambda) \hat{\mathbf{j}}-(1+2 \lambda) \hat{\mathbf{k}} \end{aligned}$ $\because$ Projection of vectord on a is $=\sqrt{\frac{2}{3}}$ $\begin{aligned} & \Rightarrow \quad \frac{\mathbf{d} \cdot \mathbf{a}}{|\mathbf{a}|}=\sqrt{\frac{2}{3}} \ & \Rightarrow \quad \frac{2(1+\lambda)-(2+\lambda)-(1+2 \lambda)}{\sqrt{4+1+1}}=\sqrt{\frac{2}{3}} \ & \Rightarrow \quad 2+2 \lambda-2-\lambda-1-2 \lambda=2 \ & \Rightarrow \quad \lambda=-3 \end{aligned}$ So, required vector $\mathbf{d}=-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ Hence, option (a) is correct.

Given three vectors \(\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}},…

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