Cross Product Calculator (3D Vectors)

Calculate the 3D cross product of two vectors: perpendicular result, magnitude, and parallelogram area. Includes step-by-step component derivations.

Inputs

Results

\begin{aligned} c_1 &= a_2 b_3 - a_3 b_2 \\ &= (2)(6) - (3)(5) \\ &= ? \end{aligned}
\begin{aligned} c_2 &= a_3 b_1 - a_1 b_3 \\ &= (3)(4) - (1)(6) \\ &= ? \end{aligned}
\begin{aligned} c_3 &= a_1 b_2 - a_2 b_1 \\ &= (1)(5) - (2)(4) \\ &= ? \end{aligned}
\begin{aligned} |\mathbf{c}| &= \sqrt{c_1^2 + c_2^2 + c_3^2} \\ &= \sqrt{(?)^2 + (?)^2 + (?)^2} \\ &= ? \end{aligned}
\begin{aligned} A &= |\mathbf{a} \times \mathbf{b}| \\ &= ? \end{aligned}

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What is the cross product?

The cross product of two three-dimensional vectors a and b is a third vector c = a × b that is perpendicular to both, with magnitude |a||b|sin θ, where θ is the angle between them.

Formula

The cross product is defined via the 3×3 determinant:

a×b=∣ijka1a2a3b1b2b3∣=(a2b3−a3b2) i−(a1b3−a3b1) j+(a1b2−a2b1) k\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2 b_3 - a_3 b_2)\,\mathbf{i} - (a_1 b_3 - a_3 b_1)\,\mathbf{j} + (a_1 b_2 - a_2 b_1)\,\mathbf{k}

Expanding the cofactors gives three scalar components:

c1=a2b3−a3b2,c2=a3b1−a1b3,c3=a1b2−a2b1c_1 = a_2 b_3 - a_3 b_2, \quad c_2 = a_3 b_1 - a_1 b_3, \quad c_3 = a_1 b_2 - a_2 b_1

The magnitude (and the area of the parallelogram spanned by a and b) is:

∣a×b∣=c12+c22+c32|\mathbf{a} \times \mathbf{b}| = \sqrt{c_1^2 + c_2^2 + c_3^2}

The right-hand rule

The direction of a × b follows the right-hand rule: with the fingers of the right hand pointing along a and curling toward b through the smaller angle, the extended thumb points in the direction of a × b. The cross product is therefore anti-commutative: b × a = −(a × b) — reversing the order flips the direction of the result.

Worked example

Let a = (1, 2, 3) and b = (4, 5, 6).

c1=(2)(6)−(3)(5)=12−15=−3c_1 = (2)(6) - (3)(5) = 12 - 15 = -3 c2=(3)(4)−(1)(6)=12−6=6c_2 = (3)(4) - (1)(6) = 12 - 6 = 6 c3=(1)(5)−(2)(4)=5−8=−3c_3 = (1)(5) - (2)(4) = 5 - 8 = -3

So a × b = (−3, 6, −3). The magnitude is:

(−3)2+62+(−3)2=9+36+9=54=36≈7.348\sqrt{(-3)^2 + 6^2 + (-3)^2} = \sqrt{9 + 36 + 9} = \sqrt{54} = 3\sqrt{6} \approx 7.348

This is also the area of the parallelogram with sides a and b.

Geometric interpretation

The magnitude |a × b| gives the area of the parallelogram whose adjacent sides are a and b. Halving it gives the area of the triangle formed by the two vectors — a result used in 3D geometry and computer graphics to compute surface area from vertex coordinates.

When the two vectors are parallel, sin θ = 0, so a × b = 0. This is a useful collinearity test: if the cross product of two edge vectors is the zero vector, the points are collinear.

Why only 3D (and 7D)?

The standard cross product requires exactly three dimensions to produce a single vector perpendicular to two given vectors using the determinant method. In 2D there is no third axis; in 4D and higher, the set of vectors perpendicular to two given vectors forms a subspace with more than one dimension, so there is no unique "perpendicular vector." A generalized cross product does exist in 7 dimensions via octonion algebra, but the 3D version covers virtually all practical applications in physics, engineering, and graphics.

Applications

The cross product appears wherever a perpendicular axis or area is needed:

  • Torque: τ = r × F, where r is the lever arm and F is the applied force.
  • Angular momentum: L = r × p.
  • Surface normals: in 3D rendering, the normal to a triangular polygon is the cross product of two edge vectors.
  • Magnetic force: F = q(v × B) — the Lorentz force on a moving charge.
  • Area of a triangle: half the magnitude of the cross product of two edge vectors.

Frequently Asked Questions (FAQ)

What does the cross product represent geometrically?

The cross product a × b produces a vector perpendicular to both a and b. Its direction is given by the right-hand rule and its magnitude equals |a||b|sin θ, where θ is the angle between a and b. Geometrically, the magnitude is the area of the parallelogram with sides a and b. If a × b = 0, the vectors are parallel (or one is zero).

What is the right-hand rule?

With the fingers of the right hand pointing along a and curling toward b through the smaller angle, the extended thumb points in the direction of a × b. The cross product is anti-commutative: b × a = −(a × b), so reversing the order flips the direction of the result.

Why is the cross product defined only in 3D (and 7D)?

The standard cross product requires exactly three dimensions to produce a vector perpendicular to two given vectors using the determinant formula. In 2D there is no third axis; in 4D and higher, there are infinitely many directions perpendicular to two vectors. A generalization exists in 7D using octonion algebra, but the 3D cross product is by far the most commonly used form in physics, engineering, and computer graphics.

When is the cross product zero?

a × b = 0 when the two vectors are parallel (including antiparallel) or when either vector is the zero vector. In this case sin θ = 0, so the magnitude |a||b|sin θ = 0. Physically, parallel vectors span no area, so the parallelogram degenerates to a line.

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