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Linear transformations in three dimensions

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

F1 used 2 × 2 matrices for transformations of the plane. In F3 the same ideas extend to 3 × 3 matrices acting on column vectors in three dimensions: reflections in planes, rotations about the axes, enlargements and stretches.

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Linear transformations

A transformation T is linear if T(a + b) = T(a) + T(b) and T(λa) = λT(a) for all vectors and scalars. Every linear transformation can be written as a matrix, and it always maps the origin to itself (so a translation is not linear).

The columns of the matrix are the images of i, j and k. To find a matrix, ask where each unit vector goes.

Standard 3D matrices

The specification does not list particular 3D transformations, so an exam question will usually give the matrix or describe the transformation. These are worth recognising. Rotation through θ about an axis is anticlockwise when looking from the positive end of the axis towards O:

TransformationMatrix (rows)
Rotation θ about the x-axis(1, 0, 0), (0, cos θ, -sin θ), (0, sin θ, cos θ)
Rotation θ about the y-axis(cos θ, 0, sin θ), (0, 1, 0), (-sin θ, 0, cos θ)
Rotation θ about the z-axis(cos θ, -sin θ, 0), (sin θ, cos θ, 0), (0, 0, 1)
Reflection in the plane x = 0(-1, 0, 0), (0, 1, 0), (0, 0, 1)
Reflection in the plane y = 0(1, 0, 0), (0, -1, 0), (0, 0, 1)
Reflection in the plane z = 0(1, 0, 0), (0, 1, 0), (0, 0, -1)
Enlargement, scale factor k, centre O(k, 0, 0), (0, k, 0), (0, 0, k)

Combining transformations

If A is applied first and then B, the single matrix is BA (the first transformation goes next to the vector: BAx = B(Ax)). Matrix multiplication is not commutative, so the order matters.

The determinant of a 3 × 3 matrix is the volume scale factor; a negative determinant means orientation is reversed (as in a reflection).

Worked example

Find the image of (1, 1, 1) under a reflection in the plane x = 0 followed by a rotation of 90° about the x-axis.

  1. Reflection: (1, 1, 1) maps to (-1, 1, 1)
  2. Rotation 90° about x: (x, y, z) maps to (x, -z, y)
  3. (-1, 1, 1) maps to (-1, -1, 1)

Answer: (-1, -1, 1)

Key idea: Columns are the images of i, j, k. A then B is the matrix BA. Linear maps fix the origin. det = volume scale factor.

Check yourself

  1. The point (1, 2, 3) is rotated 90° anticlockwise about the z-axis. Find the x-coordinate of its image.
    Show answer

    -2. The matrix has rows (0, -1, 0), (1, 0, 0), (0, 0, 1), so (1, 2, 3) maps to (-2, 1, 3).

  2. Which matrix represents reflection in the plane z = 0?
    • Rows (-1, 0, 0), (0, -1, 0), (0, 0, 1)
    • Rows (-1, 0, 0), (0, 1, 0), (0, 0, 1)
    • Rows (1, 0, 0), (0, 1, 0), (0, 0, -1)
    • Rows (1, 0, 0), (0, 1, 0), (0, 0, 0)
    Show answer

    Rows (1, 0, 0), (0, 1, 0), (0, 0, -1). Points in the plane z = 0 stay fixed, and the z-coordinate changes sign.

  3. Transformation A is applied, then transformation B. Which matrix represents the combined transformation?
    • BA
    • AB
    • A + B
    • A-1B
    Show answer

    BA. BAx = B(Ax): A acts on x first.

  4. A reflection in the plane x = 0 is followed by a rotation of 90° about the x-axis. Find the y-coordinate of the image of (1, 1, 1).
    Show answer

    -1. The reflection gives (-1, 1, 1). The rotation maps (x, y, z) to (x, -z, y), giving (-1, -1, 1).

  5. What are the columns of a 3 × 3 transformation matrix?
    • The images of the axes' negative ends
    • The images of i, j and k
    • The eigenvalues
    • The coefficients of a plane
    Show answer

    The images of i, j and k. M times (1, 0, 0) picks out the first column, and so on.

  6. A solid of volume 2 cm3 is enlarged with scale factor 3, centre O. Find the volume of the image, in cm3.
    Show answer

    54 cm3. The matrix is 3I, with determinant 27, so the volume becomes 2 × 27 = 54 cm3.

  7. Which transformation of three-dimensional space is not linear?
    • A rotation about the z-axis
    • A translation by (1, 0, 0)
    • A reflection in the plane y = 0
    • An enlargement with centre O
    Show answer

    A translation by (1, 0, 0). A linear transformation maps the origin to itself; a translation moves it.

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Last updated 8 November 2026. Spotted a mistake? Tell us.