Let \(\vec{a}=2\overset{^}{i}−7\overset{^}{j}+5\overset{^}{k},\vec{b}=\overset{^}{i}+\overset{^}{k}\) and \(\vec{c}=\ove…

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

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