# Definition:Right-Hand Rule

## Definition

### Axial Vector

Let $\mathbf V$ be an axial vector acting with respect to an axis of rotation $R$.

Consider a right hand with its fingers curled round $R$ so that the fingers are pointed in the direction of rotation of $\mathbf V$ around $R$.

The **right-hand rule** is the convention that the direction of $\mathbf V$ is the direction in which the thumb is pointing:

### Cartesian 3-Space

A Cartesian $3$-Space is defined as being **right-handed** if it has the following property:

Let a **right hand** be placed such that:

- the thumb and index finger are at right-angles to each other
- the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm
- the thumb points along the $x$-axis in the positive direction
- the index finger points along the $y$-axis in the positive direction.

Then the $3$rd finger is pointed along the $z$-axis in the positive direction.

### Vector Cross Product

Let a **right hand** be placed such that:

- the $3$rd finger is at right-angles to both the thumb and index finger, upwards from the palm
- the thumb points along the direction of $\mathbf a$
- the index finger points along the direction of $\mathbf b$.

Then the $3$rd finger is pointed along the direction of $\mathbf a \times \mathbf b$.

## Also known as

The **right-hand rule** can also be seen rendered unhyphenated: **right hand rule**.