# vectors

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dot products, scalar products and inner products...

are they same thing?

No.

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what is difference?

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The scalar product, inner product and dot product are the same thing over a Euclidean space.

More generally, a vector space (over a field) with an inner product generalises the notion of the dot product on a Euclidean space.

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why different nouns?

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why different nouns?

Great question. I can't say I really know why.

Dot comes from the notation. Scalar as the result is a scalar (element of the underlying field). Inner I am not sure why.

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It can be a bit hard to chase down the origin of terms in mathematics.

My guess: Given some N×M matrix A and an R×S matrix B, the product AB only makes sense mathematically if R=M. In that case, AB is the N×S product of an N×M and an M×S matrix. The inner dimensions, M in this case must match but do not play a part in the dimensionality of the product.

An n-vector can be interpreted as being a 1×n matrix (a row vector) or an n×1 matrix (a column matrix). There are two ways in matrix theory to represent the product of two vectors: as the product of a row vector and a column vector (1×n × n×1), or as the product of a column vector and a row vector (n×1 × 1×n). The former is the "inner" product (the n's match up on the inside) while the latter is the "outer" product (the n's match up on the outside). For example, using three vectors,

$\bmatrix a & b & c\endbmatrix \bmatrix e \\ f \\ g \endbmatrix = ae+bf+cg$

$\bmatrix a \\ b \\ c\endbmatrix \bmatrix e & f & g \endbmatrix = \bmatrix ae & af & ag \\ be & bf & bg \\ ce & cf & cg\endbmatrix$

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