# Inner product and dot product

Could someone explain me the difference between the inner product and the dot product?

Thanks all

mathman
There really isn't any. The terms are used more or less interchangeably. For 2 or 3 dimensions dot product is usually used, while for higher dimensions and Hilbert space, inner product is usually used, but they mean the same thing.

HallsofIvy
Homework Helper
There is, in a sense, a technical difference. The "dot product" is defined in any $R^n$ (or $C^n$ while an "inner product" is any function from a pair of vectors, in any vector space, to the real numbers, satisfying
1) <u, v>= <v, u> (or, if the vector space is over the complex numbers, the complex conjugate of <v, u>.
2) <u+ v, w>= <u, w>+ <v, w>.
3) <ku, v>= k<u, v> for any number k.

Of course, any dot product is an inner product.

Going the other way, if we choose an orthonormal basis for the vector space, $\{v_1, v_2, v_3, \cdot\cdot\cdot, v_n\}$ any vectors u, and v, can be written as a linear combination of them, and then write the inner product as a dot product of those coefficients. The theoretical "meat" of the Gram-Schmidt procudure (that such an orthonormal basis exists) is that every inner product is a dot product- in some basis.