Concept

Normal vector

A vector perpendicular to a surface or a plane at a point, used to say which way the surface faces there. Under a linear map it is carried by the inverse transpose rather than by the map itself, which is why the normal of a stretched surface is not the stretched normal.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A frame carried round a Möbius band. A flat rectangle whose ends are about to be joined, with a pair of arrows carried along it — one along the band and one across it — and the sign of the frame at each station.

Orientation is a sign

Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.

topology · Orientability

Perpendicular to both, and as long as their parallelogram. The vectors (2, 0.4, 0.2) and (0.6, 1.8, 0.3), the parallelogram they span, and their cross product (−0.24, −0.48, 3.36), drawn perpendicular to both with a length of 3.403, which is the parallelogram's area.

A plane disguised as an arrow

The cross product of two arrows is an arrow perpendicular to both, as long as the area of their parallelogram. Reflect everything in a mirror and it points the wrong way, because it was never an arrow: it is a plane, written as the one direction a plane in three dimensions leaves over.

algebra · Inner product

Named alongside it

The objects these essays reach for when they reach for this one.

Determinant Orientation Cauchy schwarz Cross product Frame Handedness Inner product Intrinsic property Matrix Orientability Orthogonality Transport

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