Let $\vec{a}=-\hat{i}-\hat{k}, \vec{b}=-\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}+2 \hat{j}+3 \hat{k}$ b

Question

Let $\vec{a}=-\hat{i}-\hat{k}, \vec{b}=-\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}+2 \hat{j}+3 \hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} 	imes \vec{b}=\vec{c} 	imes \vec{b}$ and $\vec{r} \cdot \vec{a}=0$, then the value of $\vec{r} \cdot \vec{b}$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

(9) : Given $\vec{r} 	imes \vec{b}=\vec{c} 	imes \vec{b}$\ $\Rightarrow(\vec{r}-\vec{c}) 	imes \vec{b}=0$\ Thus, $\vec{r}-\vec{c}$ is parallel to $\vec{b}$\ $\vec{r}-\vec{c}=t \vec{b} \Rightarrow \vec{r}=\vec{c}+t \vec{b}, t$ is a scalar\ $(\vec{c}+t \vec{b}) \cdot \vec{a}=0$\ $\Rightarrow((\hat{i}+2 \hat{j}+3 \hat{k})+t(-\hat{i}+\hat{j})) \cdot(-\hat{i}-\hat{k})=0$\ $\Rightarrow \quad t-4=0 \quad 	herefore t=4$.\