If we have two vectors $w$ and $v$ (members of $\mathbb{C^n})$ I understand that
$v \cdot w = v^*w$.
But what happens if we have one real vector and one complex, say $v = \begin{bmatrix} 2 \\ -5 \\ 1 \end{bmatrix}$ and $w = \begin{bmatrix} i \\ 2+i \\ 1 \end{bmatrix}$
Taking the dot product in the usual way I would get $2i-5(2+i)+1 = -9-3i$
If I instead take the conjugate transpose of $w$ I would get
$[-i, 2-i, 1] \cdot \begin{bmatrix} 2 \\ -5 \\ 1 \end{bmatrix} = -9+3i$. What is correct?