Use vectors to prove geometric results in two dimensions (other than those listed in Unit 2 Topic 3) and in three dimensions. Specialist Mathematics 2025 v1.4

In parallelogram \(OABC\), the position vectors of \(A\) and \(C\) relative to \(O\) are \(\mathbf{a}\) and \(\mathbf{c}\) respectively. What is the position vector of the midpoint of diagonal \(OB\)?

Use vectors to prove that the four points \(A\), \(B\), \(C\) and \(D\) form a parallelogram, and determine the coordinates of the midpoint of diagonal \(AC\).

A tetrahedron \(ABCD\) has vertices \(A\), \(B\), \(C\), and \(D\). Let \(M\) be the midpoint of edge \(AB\), \(N\) be the midpoint of edge \(CD\), \(P\) be the midpoint of edge \(AC\), and \(Q\) be the midpoint of edge \(BD\). Use vectors \(\vec{AB} = \mathbf{a}\), \(\vec{AC} = \mathbf{b}\), and \(\vec{AD} = \mathbf{c}\) to prove that the line segments \(MN\) and \(PQ\) bisect each other.

A tetrahedron \(OABC\) has vertices at the origin \(O\) and at points \(A\), \(B\), and \(C\) defined by position vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) respectively. Let \(M\) be the midpoint of edge \(AB\), \(N\) be the midpoint of edge \(OC\), and \(P\) be the point on edge \(BC\) such that \(BP:PC = 2:1\). Prove that points \(M\), \(N\), and \(P\) are collinear.

A tetrahedron \(OABC\) has vertices at the origin \(O\) and points \(A\), \(B\), and \(C\) defined by position vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) respectively. The midpoint of edge \(OA\) is \(M\), the midpoint of edge \(BC\) is \(N\), the midpoint of edge \(OB\) is \(P\), and the midpoint of edge \(AC\) is \(Q\). a) Use vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) to express the vector \(\overrightarrow{MN}\). (1 mark) b) Use vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) to express the vector \(\overrightarrow{PQ}\). (1 mark) c) Hence prove that the line segment \(MN\) is parallel to the line segment \(PQ\). (1 mark)