Quaternion

A quaternion is an extension of complex numbers used to represent rotations and 3D orientation.

Definition §

A quaternion has the form

where and the imaginary units satisfy

The multiplication rules are

and reversing the order changes the sign:

This means quaternion multiplication is not commutative.

Scalar and Vector Parts §

We can write

where the scalar part is and the vector part is

So

in shorthand.

Conjugate, Norm, and Inverse §

The conjugate of

is

The norm is

and

If , then the inverse is

Unit Quaternion §

A unit quaternion satisfies

Unit quaternions are especially important because they represent rotations in 3D.

Quaternion Multiplication §

If

then

This combines the dot product and cross product into one algebraic rule.

Rotations in 3D §

To rotate by angle around a unit axis , use the unit quaternion

A vector is written as the pure quaternion

The rotated vector is obtained by

Why Quaternions Are Useful §

• They avoid gimbal lock that can happen with Euler angles
• They are more compact than rotation matrices
• They are numerically stable for interpolation
• They are widely used in computer graphics, robotics, and physics