NJC Vectors Tutorial Q2

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. The two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy the equation \(\mathbf{a}=\left( \mathbf{a}\cdot \mathbf{b} \right)\mathbf{b}\).

1. State a geometrical relationship between \(\mathbf{a}\) and \(\mathbf{b}\) and determine the value of \(\left| \mathbf{b} \right|\).[3]

A third point \(C\) has position vector \(\mathbf{c}\). It is further given that \(\mathbf{a}\cdot \mathbf{b}=-2\) and \(\mathbf{c}\cdot \left( \mathbf{c}+\mathbf{b} \right)=3\).

2. Determine if \(AC\) is perpendicular to \(BC\).[3]
3. Show that the area of triangle \(ABC\) can be expressed as \(k\left| \mathbf{b}\times \mathbf{c} \right|\), where \(k\) is a constant to be determined.[2]

Another point \(D\) lies on \(CA\) produced such that \(CD=4CA\).

4. Given that \(\overrightarrow{OA}=\mathbf{i}-2\mathbf{k}\) and \(\overrightarrow{OC}=3\mathbf{j}+4\mathbf{k}\), find the position vector of \(D\).[2]