Difference between Inner Product and Dot Product

[Taturana]

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]

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.