Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k

Question

Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} 	imes \vec{a}=\vec{c} 	imes \vec{a}$ and $\vec{r} \cdot \vec{b}=0$, then $\vec{r} \cdot \vec{a}$ is equal to $\_\_\_\_$ .\

Professor PC Thomas Classes · Answer

(12) : Given, $\vec{r} 	imes \vec{a}=\vec{c} 	imes \vec{a}$\ $\Rightarrow(\vec{r}-\vec{c}) 	imes \vec{a}=\overrightarrow{0}$\ $\Rightarrow \vec{r}-\vec{c}=\lambda \vec{a} \Rightarrow \vec{r}=\vec{c}+\lambda \vec{a}$\ $\Rightarrow \vec{r} \cdot \vec{b}=\vec{c} \cdot \vec{b}+\lambda \vec{a} \cdot \vec{b}$\ $\Rightarrow 0=\vec{c} \cdot \vec{b}+\lambda \vec{a} \cdot \vec{b} \Rightarrow \lambda=\frac{-\vec{c} \cdot \vec{b}}{\vec{a} \cdot \vec{b}}=\frac{-2}{-1}=2$\ $	herefore \vec{r} \cdot \vec{a}=\vec{c} \cdot \vec{a}+\lambda \vec{a} \cdot \vec{a}=0+2(6)=12$\