Rotation is linear

This series starts from 2D rotation matrices and takes us through the developments we have made to understand and describe rotations. It covers imaginary numbers, intrinsic and extrinsic rotations, Euler angles, quaternions, geometric algebra and Lie theory.

Rotation is linear and can be represented by a matrix.

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0. Establishing fundamental facts about rotations

First let's see that rotations are linear transformations.

0.1. Add-then-rotate = Rotate-then-add

• Sum of two rotated vectors is the same as rotating the sum of the vectors. In other words

Add then rotate ends up here.

Rotate and then add ends up at the same point.

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0.2. Scale-then-rotate = Rotate-then-scale

Now let's see linearity in action for scaling:

• Scaling a rotated vector is the same as rotating the scaled vector. In other words:

• Scale then rotate ends up here.
• Rotate then scale also ends up at the same point

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These two properties establish that rotations are linear transformations. Thus we can represent any rotation as a matrix. That is where is a matrix and is the vector to be rotated.

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1. Rotation preserves length and angle between vectors

Also note that a rotation preserves the angles between two vectors. In other words, the dot product of two rotated vectors is the same as the dot product of the original vectors:

This implies that where is the identity matrix. Thus, the inverse of a rotation matrix is its transpose. It also reveals one more fact:

The columns of a rotation matrix are orthonormal.

It is important to understand how rotation transforms space. The visual aid below will help with it:

• Define angle of rotation.
• View how it transforms a grid.
• First and second basis vectors.
• Animate theta

Note that all the above arguments are valid for rotations in any dimension and basis.

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2D rotations →