Let \(\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}\) be th…

Let \(\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that \(\vec{r} \times \vec{a}=\vec{c} \times \vec{a}\) and \(\vec{r} \cdot \vec{b}=0\), then \(|\vec{r}|\) is equal to:

[JEE Main 2023, 1 Feb (Shift 2)]