How to exploit bilinearity on the following expression - dot product

Question

I struggle to apply the concept of bilinearity to follow along with simplifying the following expression bellow.

\begin{align}\langle x-\pi_{U}(x),\space b \rangle \space= \space0\iff\langle x-\lambda,\space b\rangle = 0\space\space(1)\end{align}

where $\pi_{U}(x)=\lambda b$ can be simplified by applying bilinearity to this:

\begin{align}\langle x,b\rangle-\lambda\langle b,b\rangle =0\space\space(2)\end{align}

From the book mathematics for machine learning page 83. The book is publicly available

Can you explain the steps between these expressions?

Answer 1

Turns out it's really simple.

$\langle x-\lambda b, \space b \rangle = \langle x,b \rangle + \langle -\lambda b,\space b\rangle =\langle x, b\rangle-\lambda\langle b, b\rangle$

Found out that the lambda can be moved in and out which added some intuition for me. At first I thought I have to combine the two equations.