Vector Equation of a Line (2D & 3D)

The vector equation of a line is a core part of the Vectors topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-02). This subtopic uses the direction vector to write a straight line as \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\) in both two and three dimensions, and expresses the line through two points as \(\mathbf{r} = \mathbf{p} + \lambda(\mathbf{q}-\mathbf{p})\).

Students and teachers will cover how to test whether two lines are parallel, whether a point lies on a line, whether three points are collinear, how to find the point of intersection of two lines, and how to identify skew lines in three dimensions — core vector techniques in the HSC Extension 2 course.