Cross Product

The cross product of two vectors in space, a × b, is a vector perpendicular to both, with length |a||b| sin θ equal to the area of the parallelogram they span, and direction given by the right-hand rule. In brief: in coordinates, (1, 2, 3) × (4, 5, 6) = (2 · 6 − 3 · 5, 3 · 4 − 1 · 6, 1 · 5 − 2 · 4) = (−3, 6, −3). Its length √54 ≈ 7.35 is the area of the parallelogram built on the two vectors, and (a × b) · a = 0.

The cross product (vector product) exists only in three dimensions. It produces normals to planes, areas of triangles and parallelograms, and volumes of boxes. This guide gives the coordinate formula, the geometric meaning, algebraic properties, the scalar triple product, and applications, with worked examples and a table of results.

How do you compute a cross product?

For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃): a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). A memory aid is the symbolic determinant with the rows (i, j, k), (a₁, a₂, a₃), (b₁, b₂, b₃), expanded along the first row. For the basis vectors: i × j = k, j × k = i, k × i = j.

Example 1: a standard computation

a = (2, 1, 0), b = (1, 3, 4): a × b = (1 · 4 − 0 · 3, 0 · 1 − 2 · 4, 2 · 3 − 1 · 1) = (4, −8, 5). Check perpendicularity: (4, −8, 5) · (2, 1, 0) = 8 − 8 = 0 and (4, −8, 5) · (1, 3, 4) = 4 − 24 + 20 = 0 ✓.

Example 2: vectors in a plane

For a = (3, 0, 0) and b = (0, 2, 0), a × b = (0, 0, 6): the area of the 3 × 2 rectangle is 6, and the direction is the positive z-axis.

What does the cross product mean geometrically?

The vector a × b has three properties: it is perpendicular to both a and b; its length is |a||b| sin θ, where θ is the angle between them; and a, b, a × b form a right-handed system: point the fingers of the right hand along a, curl them towards b, and the thumb shows a × b. Parallel vectors give the zero vector, since sin 0° = sin 180° = 0.

What are its algebraic properties?

• Anticommutative: b × a = −(a × b).
• a × a = 0.
• Distributive: a × (b + c) = a × b + a × c.
• Scalars: (ka) × b = k(a × b).
• Not associative: (i × i) × j = 0 but i × (i × j) = i × k = −j.
• Lagrange’s identity: |a × b|² = |a|²|b|² − (a · b)².

Example 3: Lagrange’s identity

For a = (1, 2, 3), b = (4, 5, 6): |a|² = 14, |b|² = 77, a · b = 32. 14 · 77 − 1024 = 1078 − 1024 = 54 = |a × b|² ✓.

How do you find areas with the cross product?

The area of the parallelogram on a and b is |a × b|. The area of the triangle ABC is (1/2)|AB × AC|. Four points are coplanar when the vectors between them have a zero triple product.

Example 4: the area of a triangle in space

A(1, 0, 0), B(0, 2, 0), C(0, 0, 3). AB = (−1, 2, 0), AC = (−1, 0, 3). AB × AC = (2 · 3 − 0, 0 · (−1) − (−1) · 3, (−1) · 0 − 2 · (−1)) = (6, 3, 2). |.| = 7, so the area is 3.5.

Example 5: a normal to a plane

The plane through A, B, C of Example 4 has the normal vector (6, 3, 2), and its equation is 6x + 3y + 2z = 6 (the point A fits: 6 · 1 = 6).

What is the scalar triple product?

The number [a, b, c] = (a × b) · c equals the 3 × 3 determinant of the three vectors written as rows. Its absolute value is the volume of the parallelepiped on a, b, c; one sixth of it is the volume of the tetrahedron. It is zero exactly when the three vectors lie in one plane.

Example 6: a volume

a = (1, 0, 0), b = (1, 2, 0), c = (1, 1, 3). a × b = (0, 0, 2), so (a × b) · c = 6. The parallelepiped has the volume 6 and the tetrahedron has the volume 1.

Example 7: coplanar points

A(1, 1, 1), B(2, 3, 4), C(0, −1, −2), D(3, 5, 7). AB = (1, 2, 3), AC = (−1, −2, −3) = −AB, so AB × AC = 0 and the four points are coplanar (in fact A, B, C are collinear).

Table of examples

Where does the cross product come from?

The product appeared in Hamilton’s quaternions (the vector part of the product of two pure quaternions) and in Grassmann’s exterior algebra in the 1840s. Gibbs and Heaviside isolated it in the 1880s as an independent operation of vector analysis, and the notation a × b is due to Gibbs. A similar product exists only in dimensions 3 and 7; in other dimensions the exterior product replaces it.

Where is it used?

In mechanics, torque is r × F and angular momentum is r × p. In electromagnetism, the magnetic force on a charge is qv × B. In computer graphics, the cross product computes surface normals. In geometry it gives the plane through three points and the shortest distance between lines.

Which mistakes are common?

Mixing up the order

a × b = −(b × a). Swapping the vectors flips the sign of the result.

Sign slip in the middle component

The middle component of the determinant expansion has a minus sign: a₃b₁ − a₁b₃. Verifying that the result is perpendicular to both vectors catches this error.

Applying it in the plane

The cross product is defined in space. For plane vectors, embed them as (a₁, a₂, 0); then the result is (0, 0, a₁b₂ − a₂b₁).

Assuming associativity

(a × b) × c and a × (b × c) differ in general. Keep the brackets.