Let \(\vec{a}=\overset{^}{i}+2\overset{^}{j}+3\overset{^}{k},\vec{b}=3\overset{^}{i}+\overset{^}{j}-\overset{^}{k}\) and…

Let \(\vec{a}=\overset{^}{i}+2\overset{^}{j}+3\overset{^}{k},\vec{b}=3\overset{^}{i}+\overset{^}{j}-\overset{^}{k}\) and \(\vec{\mathrm{c}}\) be three vectors such that \(\vec{\mathrm{c}}\) is coplanar with \(\vec{\mathrm{a}}\text{ and }\vec{\mathrm{b}}.\) If the vector \(\vec{\mathrm{c}}\) is perpendicular to \(\vec{b}\) and \(\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=5,\) then \(|\vec{\mathrm{c}}|\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]