In all introductory linear algebra texts there is a discussion on orthogonal projection.
Let $u = w_1 + w_2$, where $w_1$ is the projection of $u$ along $v$ and $w_2$ is projection of orthogonal to $v$.
We know that $w_1 = \frac{u \cdot v}{v \cdot v} v $ and $||w_1|| = ||u||\, |\cos{\theta}|$ or simply $|u \cdot v|$ if $v$ is a unit vector.
In the textbook, there is no mention of $||w_2||$ and I wonder if there is a formula for it.
$|| w_2 || = || u - \frac{u \cdot v}{v \cdot v}v || \longrightarrow $ what?