2025 HCI P2 Q4

Let the \(A\), \(B\) and \(C\) be points on the same plane with position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively and \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\). It is given that vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are unit vectors.

1. 1. By considering \(\mathbf{c}\cdot \mathbf{c}\), find the value of \(\mathbf{a}\cdot \mathbf{b}\).[3]
   2. Find the angle \(\angle AOB\).[2]
   3. Draw the position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) on a single diagram. Using your diagram, identify the type of triangle \(ABC\).[2]
2. The point \(D\) has position vector \(\mathbf{a}+\mathbf{b}\). Find the area of the quadrilateral \(ACBD\).[2]