2026 ASRJC P1 Q6

Relative to the origin \(O\), the position vectors of \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively where \(\mathbf{c}=3\mathbf{b}-\mathbf{a}\). Given that \(\mathbf{b}\) is a unit vector, \(|\mathbf{a}|=3\), \(|\mathbf{b}-\mathbf{a}|=\sqrt7\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac\pi3\),

1. find the exact shortest distance from \(C\) to the line \(AB\).[4]
2. Find the angle between \(\mathbf{b}\) and \(\overrightarrow{AC}\).[3]