Relationship Dot Product and Vector Size

Suppose we take the inner product of a vector against itself, what is the that inner product equal to?

Say:

$$ r = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} $$ $$ r*r= \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} * \begin{bmatrix} 2 , 3, 4 \end{bmatrix} = 2^2 + 3^2 + 4^2 $$ $$ r^2 = 2^2 + 3^2 + 4^2 $$

Notice that the above representation is the same as the length of vector represention if you apply a square root on both sides.

Note: The size of vector is also equal to the square root of it's inner product.