Let $\vec{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \vec{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\vec{\mathbf{c}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ be three vectors. A vector $\mathbf{v}$ in the plane of $\vec{\mathbf{a}}$ and…

Question

Let $\vec{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \vec{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\vec{\mathbf{c}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ be three vectors. A vector $\mathbf{v}$ in the plane of $\vec{\mathbf{a}}$ and $\vec{\mathbf{b}}$ whose projection of $\vec{\mathbf{c}}$ is $\frac{1}{\sqrt{3}}$, is given by

Aicharya · Accepted Answer

Let $\mathbf{v}=\mathbf{a}+\lambda \mathbf{b}$ $ \mathbf{v}=(1+\lambda) \hat{\mathbf{i}}+(1-\lambda) \hat{\mathbf{j}}+(1+\lambda) \hat{\mathbf{k}} $ Projection of $\mathbf{v}$ on $\mathbf{c}=\frac{1}{\sqrt{3}}$ $ \begin{array}{lc} \Rightarrow & \frac{\mathbf{v} \cdot \mathbf{c}}{|\mathbf{c}|}=\frac{1}{\sqrt{3}} \ \Rightarrow & \frac{(1+\lambda)-(1-\lambda)-(1+\lambda)}{\sqrt{3}}=\frac{1}{\sqrt{3}} \ \Rightarrow & 1+\lambda-1+\lambda-1-\lambda=1 \ \Rightarrow & \lambda-1=1 \Rightarrow \lambda=2 \ 	herefore & \mathbf{v}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}} \end{array} $

Let $\vec{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}},…

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