Inner Product vs Dot Product: Understanding the Difference

[Jhenrique]

A simple question: what is the difference between inner product and dot product?

 

[jgens]

The dot product is just a specific inner product on Rⁿ.

 

[HallsofIvy]

An "inner product" on a given vector space V, over the complex numbers, is any function that, to any two vectors in U, u and v, assigns the complex number, <u, v> such that
1) For any vector, v, $<v, v>\ge 0$ and $<v, v>= 0$ if and only if v= 0.
2) For any vectors, u and v, and any complex number, r, r<u, v>= <ru, v>.
3) For any vectors, u and v, $<u, v>= \overline{v, u}$.

(If V is a vector space over the real numbers, <u, v> must be real and <u, v>= <v, u>.)

The "dot product on Rⁿ" is an inner product and the converse is almost true:
If we take a basis on the vector space V, consisting of "orthonormal vectors" where "orthogonal" is defined as <u, v>= 0 and "normal" as <v, v>= 1, there is a natural isomorphism from V to Rⁿ, where n is the dimension of V, so we can write u and v as "ordered n-tuples" and the inner product on V is exactly the dot product on Rⁿ.

 

[Jhenrique]

I understood. But, by the way, if there is a product between vectors involving the modulus and the sine of the angle formed and can result or a scalar or a vector (exterior product and cross product), so, similarly, no exist a prodcut between vectors involving the modulus and the cossine of the angle formed that could result or a scalar or a vector too?