2013 ACJC P2 Q1

Referred to the origin \(O\), the points \(A\) and \(B\) are such that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\), where \(\left| \mathbf{a} \right|=2\) and \(\mathbf{b}\) is a unit vector.

The midpoint of \(OA\) is \(M\), and the point \(L\) on \(AB\)  is such that \(AL:LB=1:2\).

1. Give the geometrical meanings of
   1. \(\mathbf{b}\cdot \mathbf{a}\),[1]
   2. \(\left| \mathbf{b} \times \mathbf{a} \right|\).[1]
2. Find \(\overrightarrow{OL}\) in terms of \(a\) and \(b\). Hence find the area of triangle \(OAL\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
3. Given that \(LM\) is perpendicular to \(AB\) , show that
   \(\mathbf{a}\cdot \mathbf{b}=k\)
   where \(k\) is a constant to be determined.