Inner Product
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( ยท ) by the inner product, i.e.
In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with
or, more commonly, using the absolute value, with
The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces and spanned by the vectors and correspondingly.
Read more about this topic: Angle
Famous quotes containing the word product:
“The history is always the same the product is always different and the history interests more than the product. More, that is, more. Yes. But if the product was not different the history which is the same would not be more interesting.”
—Gertrude Stein (1874–1946)
“The product of mental labor—science—always stands far below its value, because the labor-time necessary to reproduce it has no relation at all to the labor-time required for its original production.”
—Karl Marx (1818–1883)