The Cross Product of Two Vectors
Simple Explanation
The cross product (or vector product) combines two vectors into a NEW vector, perpendicular to both of the original vectors: a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁).
Why Do We Need It?
Unlike the dot product (which gives a number), the cross product gives a genuinely new vector — essential whenever a direction perpendicular to two known directions is needed, such as a surface's normal vector.
See It
Two arrows from a common origin along the x-axis and y-axis directions, with a third dashed arrow pointing straight up representing their cross product
Formula
The Cross Product
a × b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)
The cross (vector) product of two vectors produces a NEW vector, perpendicular to both original vectors.
- a = (a₁,a₂,a₃), b = (b₁,b₂,b₃)
- — the two vectors being combined
- a × b
- — the resulting vector, perpendicular to both a and b
When to use it: Whenever a vector perpendicular to two given vectors is needed.
Worked Example
Find the cross product of two vectors
Find a×b for a=(1,0,0) and b=(0,1,0).
Why Does This Work?
This particular result — that the vector along x, crossed with the vector along y, gives exactly the vector along z — is precisely the geometric meaning of the cross product: it always produces a vector perpendicular to both original vectors, following a consistent "right-hand rule" direction.
Real-Life Example
Finding the direction perpendicular to a surface
A computer graphics program knows two direction vectors lying flat within a surface (like a 3D model's face) and needs the surface's normal direction for lighting calculations.
The cross product of the two in-surface vectors gives exactly the perpendicular (normal) direction needed.
Practice
Find the z-component of a×b for a=(2,1,0) and b=(1,3,0).
HardCommon mistake
Mixing up the cyclic index pattern of the cross product formula (e.g. computing a₁b₃−a₃b₁ for the wrong component) — each component follows a specific cyclic pattern that must be applied exactly.
Quick Review
- a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁).
- The result is a NEW vector, perpendicular to both a and b.
- i×j=k is the classic example of this pattern.