# 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 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 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.