Vectors - Magnitude of a Vector

Given two vectors \(\mathbf{a}\) and \(\mathbf{b}\) such that \(|\mathbf{a}|=5\) and \(|\mathbf{b}|=4\), and that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\theta=30^\circ\), find the value of \(\mathbf{a}\cdot\mathbf{b}\) in exact form.

Another vector \(\mathbf{c}\) is such that \(\mathbf{c}=3\mathbf{a}-2\mathbf{b}\). Find the exact magnitude of \(\mathbf{c}\).