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?

 

[blue_raver22]

The inner product and dot product are two mathematical operations that are often confused with each other, but they have distinct differences. The main difference between the two is the type of vectors they operate on.

The inner product is defined as the sum of the products of the corresponding entries of two vectors. It is typically used for vectors in a complex vector space, where the vectors can have both real and imaginary components. The result of an inner product is a complex number.

On the other hand, the dot product is defined as the sum of the products of the corresponding entries of two vectors, but in this case, the vectors are in a real vector space, where the vectors only have real components. The result of a dot product is a real number.

Another important difference between the inner product and dot product is their geometric interpretations. The inner product is used to measure the angle between two vectors, while the dot product is used to measure the projection of one vector onto another.

In summary, the inner product and dot product are two distinct mathematical operations that have different applications and operate on different types of vectors. It is important to understand their differences in order to use them correctly in various mathematical and scientific contexts.