Contents

- Introduction
- The Cross Product
- Finding the Intersection of two Planes
- Finding the Angle between two Planes
- Finding the Intersection of Lines
- Calculating the Distance between a Point and a Line
- Calculating the shortest Distance between two Skew Lines
- Calculating the Distance between a Point and a Plane
- The Scalar Triple Product

### Introduction

### The Cross Product

#### What is the cross product?

**, also known as the vector product, of two vectors gives a vector perpendicular to both the vectors, in a**

*cross product**"right hand set"*. This can be very useful in defining planes etc. The cross product of two vectors is given by the following:

You can use your ** right hand** with your thumb, index finger and second finger to see in which direction the cross product of two vectors point:

- Use your index finger to represent the first vector (
**a**) - Use your second finger to represent the second vector (
**b**) - Your thumb will be pointing in the direction of the cross product of the two vectors (
**axb**)

#### Calculating the Cross Product of 2 vectors

#### Properties of the Cross product operation

*1. The Cross product is anti-commutative*The cross product is an anti-commutative operation. This means that:

*2. The Cross product of parallel vectors is zero*The Magnitude of the cross product of two vectors is proportional to the sign of the angle between them. If they are parallel the angle between them is 0, therefore since sin(0)=0, the cross product will be equal to 0: The cross product is an anti-commutative operation. This means that:

*3. Multiplication by scalars*Because you are able to describe the magnitude of a scalar-vector product in the following way:

*4. The distributive property of the cross product over vector addition*Just as scalar multiplication is distributive over scalar addition, the cross product is distributive over vector addition. What does this mean? This means that if you crossing a vector by a sum of two other vectors, you can separate the calculation into two:

### Finding the Intersection of two Planes

#### Method 1: Find two points

Let's say, for example, that you want to find the line of intersection between the following two planes:**P**:

**Q**:

#### Method 2: Crossing the normal vectors

#### Method 3: Substituting a parameter

### Finding the Angle between two Planes

### Finding the Intersection of Lines

### Calculating the Distance between a Point and a Line

### Calculating the shortest Distance between two Skew Lines

**PQ**is clearly the shortest distance, and is

**. This means that taking the**

*perpendicular to both lines***of both lines will give us the direction of the shortest distance line (**

*cross product of the direction vectors***PQ**).

**of the shortest distance line, we need to find its**

*direction***. From the vector equation of a line, you have a point on each line (the starting point).**

*magnitude***A1**and the point

**B**is equal to (

**PQ**) therefore is also equal to the shortest distance we want to find.

**A1A2**and

**A1B**are base-to-base therefore the dot product will give us the following, remembering that

**A1B**is in the same direction as the normal

**n**we previously worked out:

**|A1A2|**.

*cosθ***|A1A2|**, thus the shortest distance between the two skew lines!

*cosθ*