Angle between Lines (Edexcel A-Level Further Mathematics): Revision Notes

6.1.3 Angle between Lines

Introduction

The scalar (dot) product of two vectors is a fundamental tool in 3D geometry. It is used to:

1. Find the angle between two vectors.
2. Express the equation of a plane.
3. Calculate angles between lines, between planes, and between a line and a plane. The scalar product formula is:

$$\mathbf{a} \times \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta$$

where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

Key Applications of the Scalar Product

Angle Between Two Lines

The angle between two lines is the angle between their direction vectors.

Let $\mathbf{b}_1$ and $\mathbf{b}_2$ be the direction vectors of two lines:

$$\cos \theta = \frac{\mathbf{b}_1 \cdot \mathbf{b}_2}{|\mathbf{b}_1||\mathbf{b}_2|}$$

Equation of a Plane

The equation of a plane can be expressed as:

$$\mathbf{r} \times \mathbf{n} = k$$

where:

• $\mathbf{r}$ is the position vector of any point on the plane.
• $\mathbf{n}$ is the normal vector to the plane.
• $k$ is a constant determined by substituting a point $\mathbf{r}_0$ into $\mathbf{n} \times \mathbf{r}_0 = k$

Angle Between Two Planes

The angle between two planes is the angle between their normal vectors.

Let $\mathbf{n}_1$ and $\mathbf{n}_2$ be the normal vectors of two planes:

$$\cos \theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|}$$

Angle Between a Line and a Plane

The angle between a line and a plane is the complement of the angle between the line's direction vector and the plane's normal vector.

If $\mathbf{b}$ is the direction vector of the line and $\mathbf{n}$ is the plane's normal vector:

$$\sin \theta = \frac{\mathbf{b} \times \mathbf{n}}{|\mathbf{b}||\mathbf{n}|}$$

where $\theta$ is the angle between the line and the plane.

Worked Examples

Note Summary