2020 EJC P2 Q4

Vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that the modulus of \(\mathbf{a}\) is \(2\) and \(\mathbf{b}\) is a unit vector perpendicular to \(\mathbf{a}\).

1. State the value of \(\mathbf{a}\cdot (2\mathbf{a}\times 3\mathbf{b})\) and explain your answer briefly.[1]
2. Find the numerical area of the parallelogram with adjacent sides defined by \(2\mathbf{a}+\mathbf{b}\) and \(5\mathbf{a}-7\mathbf{b}\).[3]

A vector \(\mathbf{c}\) is such that \(\mathbf{b}\times \mathbf{c}=2\mathbf{a}\times 3\mathbf{b}\).

3. Show that \(\mathbf{c}=\lambda \mathbf{b}-6\mathbf{a}\), where \(\lambda \) is a constant.[2]
4. State the geometrical meaning of \(\left| \mathbf{b}\cdot \mathbf{c} \right|\) and find the possible values of \(\lambda \) if \(\left| \mathbf{b}\cdot \mathbf{c} \right|=5\).[2]