This is Chapter 5 – Vectors for the Year 12 Specialist Mathematics course.

This chapter extends the vector foundations developed in Year 11 into three-dimensional space. Students use vectors as both algebraic objects and geometric tools, describing position, direction and relationships in a form that can be manipulated precisely.

We begin with vectors in space and the operations used to combine them. Addition, subtraction and scalar multiplication are developed in component form, while vector algebra allows students to simplify expressions and solve geometric problems without relying solely on a diagram.

The vector between two points and tests for parallelism strengthen coordinate reasoning in three dimensions. The scalar product is then used to calculate angles and determine orthogonality, linking algebraic calculations directly to geometric relationships. Students must move fluently between component notation and geometric meaning, since the same calculation may represent a length, direction, angle or condition for perpendicularity.

Proof using vector geometry requires students to organise these ideas into clear logical arguments. Rather than assuming a relationship from a sketch, students establish it using vector equations, parallelism or scalar products. The vector product then introduces a new three-dimensional operation with applications to area and orientation.

Vectors are central to the chapters that follow, particularly lines, planes and vector calculus. They are also used extensively in physics, engineering, computer graphics, robotics and aerospace modelling, where quantities such as force, velocity, position and direction must be represented in three dimensions.

Subchapters:

• 5A: Vectors in space
• 5B: Operations with vectors in space
• 
  5C: Vector algebra
• 5D: The vector between two points
• 5E: Parallelism
• 5F: The scalar product of two vectors
• 5G: The angle between two vectors
• 5H: Proof using vector geometry
• 5I: The vector product of two vectors

By the end of this chapter, students will be able to perform vector operations in space, calculate angles and relationships, and use scalar and vector products within geometric reasoning and proof.