# Cross Product

A vector has **magnitude** (how long it is) and **direction**:

**Two vectors** can be **multiplied** using the "**Cross Product**" *(also see Dot Product)*

The Cross Product **a × b** of two vectors is **another vector** that is at right angles to both:

And it all happens in 3 dimensions!

The magnitude (length) of the cross product equals the area of a parallelogram with vectors **a** and **b** for sides:

See how it changes for different angles:

The cross product (*blue*) is:

- zero in length when vectors
**a**and**b**point in the same, or opposite, direction - reaches maximum length when vectors
**a**and**b**are at right angles

And it can point one way or the other!

So how do we calculate it?

## Calculating

#### We can calculate the Cross Product this way:

**a × b** = |**a**| |**b**| sin(θ) **n**

- |
**a**| is the magnitude (length) of vector**a** - |
**b**| is the magnitude (length) of vector**b** - θ is the angle between
**a**and**b** **n**is the unit vector at right angles to both**a**and**b**

So the **length** is: the length of **a** times the length of **b** times the sine of the angle between **a** and **b**,

Then we multiply by the vector **n** to make sure it heads in the right **direction** (at right angles to both **a** and **b**).

#### OR we can calculate it this way:

When **a** and **b** start at the origin point (0,0,0), the Cross Product will end at:

**c**_{x}= a_{y}b_{z}− a_{z}b_{y}**c**_{y}= a_{z}b_{x}− a_{x}b_{z}**c**_{z}= a_{x}b_{y}− a_{y}b_{x}

### Example: The cross product of **a** = (2,3,4) and **b** = (5,6,7)

- c
_{x}= a_{y}b_{z}− a_{z}b_{y}= 3×7 − 4×6 = −3 - c
_{y}= a_{z}b_{x}− a_{x}b_{z}= 4×5 − 2×7 = 6 - c
_{z}= a_{x}b_{y}− a_{y}b_{x}= 2×6 − 3×5 = −3

Answer: **a × b** = (−3,6,−3)

## Which Direction?

The cross product could point in the completely opposite direction and still be at right angles to the two other vectors, so we have the:

"Right Hand Rule"

With your right-hand, point your index finger along vector **a**, and point your middle finger along vector **b**: the cross product goes in the direction of your thumb.

## Dot Product

The Cross Product gives a **vector** answer, and is sometimes called the **vector product**.

But there is also the Dot Product which gives a **scalar** (ordinary number) answer, and is sometimes called the **scalar product**.

Question: What do you get when you cross an elephant with a banana?

Answer: |**elephant**| |**banana**| sin(θ) **n**