TITLE: difference between dot product and inner product
QUESTION [72 upvotes]: I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$.  And would anyone agree that an inner product is a term used when discussing the integral of the product of $2$ functions is equal to $0$?  Or is there no difference at all between a dot product and an inner product?

REPLY [6 votes]: Let...

$c\in\mathbb{C}$
$\vec{u}$, $\vec{v}\in\mathbb{R}_{m\times1}$
$A$, $B\in\mathbb{C_{m\times n}}$
Complex conjugate of $c = \overline{c} = c^*$
Absolute value of $c = \left|c\right| = \sqrt{\overline{c}c}$

Now, according to wiki...

Dot product of $\vec{u}$ & $\vec{v} = \vec{u}\cdot\vec{v} = \vec{u}^T\vec{v}$.
Inner product of $\vec{u}$ & $\vec{v} = \left<\vec{u},\vec{v}\right> = \vec{u}^T\vec{v}$.

As mentioned in the chosen answer, the inner product is sometimes used as a generalization of the dot product (& the referenced article talks about this) but, in the interest of brevity/utility, I advise you to consider them equivalent unless you have reason to believe otherwise (i.e., strange notation, explicit definitions, context, etc.).
As silly as this sounds, I tend to use $\left<\vec{u},\vec{v}\right>$ in place of $\vec{u}\cdot\vec{v}$ simply bc it is more explicit/identifiable (i.e., my brain is hardwired to associate "$\cdot$" w/ scalar multiplication) &, if my memory serves me, I picked up the habit from Dr. Strang.
P.S. To extend the ideas of dot/inner products to arbitrary real/complex matrices, checkout the Frobenius inner product...
$$
A:B = \left<A,B\right>_F = \text{tr}\left(\overline{A}^TB\right)
$$