Cross Product

1. What Is the Cross Product?

The cross product (or vector product) takes two 3D vectors and returns a new vector — not a number. That output vector is perpendicular to both of the inputs, and its length measures the area of the parallelogram they span. Unlike the dot product, which collapses two vectors into a scalar, the cross product builds a third direction out of the first two.

The cross product is defined only in three dimensions. It is the natural tool for finding surface normals, torque, angular momentum, and any “right-angle” direction in 3D space.

외적(벡터곱)은 두 3차원 벡터를 받아 새로운 벡터를 반환합니다. 그 벡터는 두 입력 벡터 모두에 수직이며, 길이는 두 벡터가 이루는 평행사변형의 넓이와 같습니다. 내적과 달리 결과가 벡터이고, 외적은 3차원에서만 정의됩니다.

2. The Formula & a Worked Example

Each output component is a small “cross” of the other two axes:

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

Worked example. For a = (2, 3, 4) and b = (5, 6, 7):

• x: a₂b₃ − a₃b₂ = (3)(7) − (4)(6) = 21 − 24 = −3
• y: a₃b₁ − a₁b₃ = (4)(5) − (2)(7) = 20 − 14 = 6
• z: a₁b₂ − a₂b₁ = (2)(6) − (3)(5) = 12 − 15 = −3

So a × b = (−3, 6, −3). You can confirm it is perpendicular to both inputs: a · (a × b) = (2)(−3) + (3)(6) + (4)(−3) = 0, and b · (a × b) = (5)(−3) + (6)(6) + (7)(−3) = 0.

각 성분은 다른 두 축의 "교차" 곱입니다: a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). 예: (2, 3, 4) × (5, 6, 7) = (−3, 6, −3). a·(a×b) = 0, b·(a×b) = 0 이므로 두 벡터 모두에 수직임을 확인할 수 있습니다.

3. The Right-Hand Rule & Parallelogram Area

The length of the cross product equals the lengths times the sine of the angle between the vectors:

|a × b| = |a| |b| sin θ

That is exactly the area of the parallelogram spanned by a and b. The direction is given by the right-hand rule: point your fingers from a toward b, and your thumb points along a × b.

Because sin θ = 0 when the vectors are parallel, the cross product of parallel vectors is the zero vector — there is no parallelogram to span.

외적의 크기는 |a × b| = |a| |b| sin θ 로, 두 벡터가 이루는 평행사변형의 넓이와 같습니다. 방향은 오른손 법칙으로 정해집니다(a에서 b로 손가락을 감으면 엄지가 a × b 방향). 평행한 벡터끼리는 sin θ = 0 이므로 외적이 영벡터입니다.

4. Key Properties

• Anti-commutative. b × a = −(a × b) — swapping the order flips the direction.
• Self cross = zero. a × a = 0, and more generally any parallel pair gives 0.
• Distributive. a × (b + c) = a × b + a × c.
• Scalar factoring. (k a) × b = k (a × b).
• Perpendicular to both. a · (a × b) = 0 and b · (a × b) = 0 always.

Note the cross product is not associative: (a × b) × c ≠ a × (b × c) in general.

반교환법칙 b × a = −(a × b), 자기외적 a × a = 0, 분배법칙 a × (b + c) = a × b + a × c, 스칼라 (k a) × b = k (a × b), 두 입력에 항상 수직. 단, 결합법칙은 성립하지 않습니다.

5. Frequently Asked Questions

Is the cross product a vector or a scalar? A vector — perpendicular to both inputs. (The dot product is the operation that returns a scalar.)

Does the cross product work in 2D? Not as a vector. The cross product is defined in 3D; in 2D people often use the scalar a₁b₂ − a₂b₁, which is the z-component alone.

Why is a × b zero when the vectors are parallel? Because |a × b| = |a||b| sin θ, and sin θ = 0 for parallel vectors — the spanned parallelogram has zero area.

외적은 벡터입니다(두 입력에 수직) — 스칼라를 반환하는 것은 내적입니다. 외적은 3차원에서 정의되며, 2D에서는 보통 z성분 a₁b₂ − a₂b₁ 만 스칼라로 사용합니다. 평행한 벡터는 sin θ = 0 이라 외적이 영벡터입니다.

Ready to practice? Drill the cross product and the rest of linear algebra on C:Vector, or review the full vector reference and the related dot product, magnitude, and angle between vectors.

실전 연습은 C:Vector에서, 전체 개념은 벡터 레퍼런스와 내적, 크기, 벡터 사이 각 문서에서 이어집니다.