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Coordinate transformations under rotation

Use the 2D rotation matrix to transform vector components under an active rotation of the vector or a passive rotation of the axes.

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  • University Physics Year 1
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Rotation matrices describe how vectors and coordinates change under a rotation. The main confusion is that there are two equally valid viewpoints:

  • Active rotation: rotate the vector in a fixed coordinate system.
  • Passive rotation: rotate the axes (change the coordinate system used to describe the same vector).

1) Active rotation (rotate the vector, axes fixed)

Let a vector have components (x,y) in the (i hat,j hat) basis. If you rotate the vector anticlockwise by an angle θ, the new components are:

(x'; y') = (cos θ, - sin θ; sin θ, cos θ) (x; y).

So:

x' = x cos θ-y sin θ, y' = x sin θ + y cos θ.

The matrix

R(θ) = (cos θ, - sin θ; sin θ, cos θ)

is the 2D rotation matrix.

2) Passive rotation (rotate the axes, vector fixed)

Now keep the vector fixed in space, but rotate the coordinate axes anticlockwise by θ. The components in the new rotated axes (x',y') are:

(x'; y') = R(-θ) (x; y) = (cos θ, sin θ; - sin θ, cos θ) (x; y).

So:

x' = x cos θ + y sin θ, y' = -x sin θ + y cos θ.
Why does the sign flip?

Rotating the axes anticlockwise by θ is equivalent to rotating the vector clockwise by θ relative to the axes, so the inverse matrix appears.

3) Key properties (good to remember)

  • R(θ) is orthogonal: R(θ)^TR(θ) = I
  • R(θ)⁻¹ = R(θ)^T = R(-θ)
  • det(R(θ)) = 1

4) Worked example

Worked example 1

Rotate a vector by a quarter turn

Problem

Rotate the vector (1,0) anticlockwise by θ = π/2.

Show full solution

  1. Write the rotation matrix

    Method

    Substitute θ = π/2 into R(θ).

    Reason

    cos π/2 = 0 and sin π/2 = 1.

    Working

    Answer

    R(π/2) = (0, -1; 1, 0).

  2. Multiply

    Method

    Apply the matrix to (1,0).

    Reason

    Multiplying by (1,0) picks out the first column.

    Working

    Answer

    (0, -1; 1, 0) (1; 0) = (0; 1).

  3. State the answer

    Working

    Answer

    (1,0) becomes (0,1): the unit vector along x turns onto y.

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