Practice question · Multiple choice
The dot product of two vectors is a number; the cross product is a vector. Why does one of them exist in every dimension while the other is peculiar to three?
Hints
- In the plane, how many directions are perpendicular to a given plane? In three dimensions? In four?
- The output of a cross product has to be a specific vector. What makes it specific?
Show the answer
A. Because only in three dimensions does a plane leave one perpendicular.
Why
The dot product's formula has nothing dimension-specific in it. The cross product is defined by a geometric demand, and in three dimensions a plane leaves exactly one perpendicular line; in four it leaves a whole plane. What generalises is the wedge product, the cross product is a three-dimensional coincidence, and physics uses it because three dimensions is where we live.
Practise Cross Product and Geometry in Space
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