I'm reading about vectors in $\mathbb{C}^n$ and my textbook defines the projection of $\vec{v}$ onto $\vec{w}$ as $$\text{proj}_\vec{w}(\vec{v})=\frac{\langle\vec{v},\vec{w}\rangle}{\lVert\vec{w}\rVert^2}\vec{w}=\langle\vec{v},\hat{w}\rangle\hat{w},$$ where $\vec{v},\vec{w}\in\mathbb{C}^n$ with $\vec{w}\ne\vec{0}$. Here, $\hat{w}$ represents the unit vector in the direction of $\vec{w}$.
How did they get from the second expression to the third?
Since we have linearity in the first argument, shouldn't the second expression simplify to $$\left\langle \frac{\vec{v}}{\lVert\vec{w}\rVert^2},\vec{w} \right\rangle\vec{w}?$$
I do not see how they arrive at the final expression from this point.
Thanks in advance!
Note that $\hat{w} = \vec{w}/\|\vec{w}\|$. (Perhaps you conflated $\hat{w}$ with $\vec{w}$ when reading the equations?)
In general, $z \langle \vec{a}, \vec{b}\rangle = \langle \vec{a}, \bar{z} \vec{b}\rangle$ for a complex scalar $z$. Since $1/\|\vec{w}\|$ is real, we have $\langle \vec{v}, \vec{w} \rangle / \|\vec{w}\| = \langle \vec{v}, \hat{w}\rangle$.