Inner Product=Dot Product always?

[Superposed_Cat]

What is the difference between a dot product and an inner product. The internet says that they are generalizations of each other. What does that even mean? Thanks for any help.

 

[UltrafastPED]

Inner product <,> is an operation defined in a vector space - if the vector space has an inner product, it is called an "inner product space".

You can measure things with your inner product: lengths and angles.

The common dot product is an example of an inner product that is algebraic; it exists in a Euclidean space, and corresponds to the Pythagorean theorem. Hence the really simple rule for evaluation when using rectangular coordinates.

Some inner products are defined by integrals ... you will run into these are your physics education progresses.

 

[Superposed_Cat]

Whats the relation between them?

 

[hilbert2]

A real inner product $<\cdot,\cdot>$ of two vectors is an operation that satisfies the following rules:

1. $<x,y>=<y,x>$
2. $<\alpha x + \beta y,z>=\alpha <x,z> + \beta <y,z>$ for any real numbers $\alpha$ and $\beta$ and vectors $x,y,z$.
3. $<x,x>\geq 0$ and $0$ only if $x=0$ (zero vector).

For example, the dot product of two vectors in ℝ³ is defined as

$<x,y>=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$. You can check that this satisfies the rules and therefore the dot product is an inner product. However, also any modified dot product defined by $<x,y>=ax_{1}y_{1}+bx_{2}y_{2}+cx_{3}y_{3}$, where $a,b,c$ are positive real constants, also satisfies the rules and is an inner product too. Therefore the usual dot product is only one example of an inner product.