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Let ak and bk be two arbitrary vectors of the
same order, and p an arbitrary scalar from the vector's basic set. Then,
the inner product
p := ab
is calculated by
calculating the sum of the element-wise products p :=
a1b1
+ a2b2
+ ... + akbk
The fact that the inner product of two vectors result in
a scalar rather than in a vector sometimes gives rise to mistakes. The
inner product is occasionally called a "scalar product", which should be
avoided in order to not mix up the inner product and the multiplication
of a vector by a scalar. Another name for the inner product is a "dot product".