The Scalar (Dot) Product

The scalar (dot) product is a central tool in the Vectors topic of NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-02). This subtopic works with the algebra of the dot product: order does not matter, \(\mathbf{a}\cdot\mathbf{b} = \mathbf{b}\cdot\mathbf{a}\); it spreads over addition, \(\mathbf{a}\cdot(\mathbf{b}+\mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c}\); and a scalar factor can be moved freely.

Students and teachers will cover why the unit vectors give \(\mathbf{i}\cdot\mathbf{i} = \mathbf{j}\cdot\mathbf{j} = \mathbf{k}\cdot\mathbf{k} = 1\) and \(\mathbf{i}\cdot\mathbf{j} = \mathbf{j}\cdot\mathbf{k} = \mathbf{k}\cdot\mathbf{i} = 0\), how this leads to the component formula, and how to find angles and test for perpendicular vectors — used throughout the HSC Extension 2 course.