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let a= 2i+k, b= i+j+k, c= 4i-3j+7k, be three vectors. find a vector r which satisfies rb=cb and r.a=0.

Let $\vec{a}=2 \hat{i}+ \hat{k}$ , $\vec{b}= \hat{i}+ \hat{j}+ \hat{k}$ and $\vec{c}=4 \hat{i}-3 \hat{j}+7 \hat{k}$. Find a vector $\vec{r} $ which satisfies $\vec{r} \cdot \vec{b}=\vec{c} \cdot \vec{b}$ and $\vec{r} \cdot \vec{a}=0$

Is is not possible to get a unique vector here? I am only able to see two equations but there are three unknowns $(\vec{r}=p\hat{i}+ q\hat{j}+ r\hat{k})$

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It is not possible to have a unique vector. If $\vec{r}$ is solution than $\alpha\vec{r}$ is also solution, for any real $\alpha$. You can try to find one for which $|\vec{r}|=1$

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    $\vec{r} \cdot \vec{b}=\vec{c} \cdot \vec{b}$ $\;\kern.6em\not\kern -.6em \implies\;$ $\alpha \vec{r} \cdot \vec{b}=\vec{c} \cdot \vec{b}$ – 2017-02-19
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    @dxiv Oops. Sorry. I mistakenly read it as $\vec{r}\cdot\vec{b}=\vec{r}\cdot\vec{c}$. I'll fix my answer – 2017-02-19