Inner product justification with an example

Question

i would need an explanation with an example on why is the inner product defined as it is and not like a dot product:

\begin{align} \text{inner pr.:}& & &\boxed{a\cdot b = a_1\overline{b_1} + a_2\overline{b_2}+...a_n\overline{b_n}}\\ \text{dot pr.:}& & &\boxed{a\cdot b = a_1b_1 + a_2b_2+...a_nb_n} \end{align}

Here it says:

For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance the dot product of a vector with itself would be an arbitrary complex number, and could be zero without the vector being the zero vector.

Can anyone provide me an example on this and show me how the definition for inner product above solves this?

Answer 1

Hint: With $u = (1+i, 0)$, what is the dot product $u\cdot u$?

Hint: With $v = (1, i)$, what is the dot product $v \cdot v$?

How do we classify all 2-d vectors such that $v \cdot v = 0$?