All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

The vector product

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

The vector product a × b of two vectors is a new vector, perpendicular to both. Its length gives the area of a parallelogram, and it is the key tool for normals to planes in the rest of F3.

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Definition

a × b = |a||b| sin θ n̂, where θ is the angle between a and b, and n̂ is the unit vector perpendicular to both, chosen by the right-hand rule (a = first finger, b = second finger, a × b = thumb).

In components, with a = (a1, a2, a3) and b = (b1, b2, b3):

a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).

This is the 'determinant' with i, j, k in the top row, a in the second and b in the third.

Properties

Areas

|a × b| = |a||b| sin θ, which is the area of the parallelogram with sides a and b.

The triangle ABC has area (1/2)|AB × AC|.

Worked example

Find the area of the triangle with vertices A(1, 0, 0), B(0, 2, 0) and C(0, 0, 3).

  1. AB = (-1, 2, 0), AC = (-1, 0, 3)
  2. AB × AC = (2×3 - 0×0, 0×(-1) - (-1)×3, (-1)×0 - 2×(-1)) = (6, 3, 2)
  3. |AB × AC| = sqrt(36 + 9 + 4) = 7

Answer: Area = 7/2

Key idea: a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1), perpendicular to a and b. |a × b| is the parallelogram area; a triangle is half. b × a = -(a × b).

Check yourself

  1. Find (i + 2j + 3k) × (4i + 5j + 6k).
    • -3i + 6j - 3k
    • 3i - 6j + 3k
    • 4i + 10j + 18k
    • -3i - 6j - 3k
    Show answer

    -3i + 6j - 3k. (2×6 - 3×5, 3×4 - 1×6, 1×5 - 2×4) = (-3, 6, -3).

  2. Find the area of the triangle with vertices A(1, 0, 0), B(0, 2, 0) and C(0, 0, 3).
    Show answer

    3.5. AB × AC = (6, 3, 2), which has length 7. The area is 7/2 = 3.5.

  3. How is b × a related to a × b?
    • b × a = -(a × b)
    • b × a = a × b
    • b × a = 0
    • b × a = |a × b|
    Show answer

    b × a = -(a × b). Swapping the order reverses the direction (right-hand rule) but keeps the length.

  4. Find the area of the parallelogram with sides a = 2i + j and b = j + 3k.
    Show answer

    7. a × b = (1×3 - 0×1, 0×0 - 2×3, 2×1 - 1×0) = (3, -6, 2), with length sqrt(9 + 36 + 4) = 7.

  5. What is k × j?
    • i
    • -i
    • -k
    • 0
    Show answer

    -i. j × k = i, so k × j = -i.

  6. Find the k-component of (3i - j + 2k) × (i + 4j - 2k).
    Show answer

    13. a1b2 - a2b1 = 3×4 - (-1)×1 = 13.

  7. Non-zero vectors a and b satisfy a × b = 0. What does this tell you?
    • The vectors are perpendicular
    • The vectors are parallel
    • The vectors have equal lengths
    • Their scalar product is zero
    Show answer

    The vectors are parallel. |a × b| = |a||b| sin θ = 0 means sin θ = 0, so θ = 0 or 180°.

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