If \(\vec{a}=\hat{i}+2 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=7 \hat{i}-3 \hat{j}+4 \hat{k}\), \(\vec{r} \tim…

If \(\vec{a}=\hat{i}+2 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=7 \hat{i}-3 \hat{j}+4 \hat{k}\), \(\vec{r} \times \vec{b}+\vec{b} \times \vec{c}=\overrightarrow{0}\) and \(\vec{r} \cdot \vec{a}=0\). Then \(\vec{r} \cdot \vec{c}\) is equal to

[JEE Main 2023, 29 Jan (Shift 2)]