Edit: Perhaps a little too much background info, for the actual question please scroll down.
I was thinking how to come up with a graphical explanation for the Gram-Schmidt orthogonalization (as it would be more intuitive for people new to the method). However, I ran intro a problem in the case where your vectors span the $\mathbb{R}^3$.
As for the $\mathbb{R}^2$ it's easy:
In order to get $v_2$ from the known vector $v$ and $w_1$, observe that $v = v_1 + v_2$ and hence $v_2 = v - v_1$. Therefore we have to find $v_1$.
Observe that the length of $v_1$ , i.e. $\|v_1\|$, equals $\|v\| \; cos \varphi$. The direction of $v_1$ is the same as the first orthogonal vector $w_1$. So $v_1 = (\|v\| \; cos \varphi) \dfrac{w_1}{\|w_1\|}$.
This looks a little like an inner product, let's write it like one.
$\begin{align} v_1 = (\|v\| \; cos \varphi) \dfrac{w_1}{\|w_1\|} = (\|v\| \; cos \varphi) \dfrac{w_1}{\|w_1\|} \dfrac{\|w_1\|}{\|w_1\|} = \dfrac{
And therefore, $v_2 = v - v_1 = v - \dfrac{
Now for $\mathbb{R}^3$, the same idea.
Because $u_3 = v_3 - y$, we have to find $y$. Well, $y = y_1 + y_2$. The length of $y$ equals $\|v_3\| cos \theta$.
Therefore, $y_1 = \|v_3\| cos \theta \; cos \varphi \dfrac{w_1}{\|w_1\|}$, and $y_2 = \|v_3\| cos \theta \; sin \varphi \dfrac{w_2}{\|w_2\|}$.
But from the normal derivation of Gram-Schmidt, $y_1$ should be $\dfrac{
So my question, how can I get to this result? Say the angle between $v_3$ and $w_1$ is $\psi$, then from the above equations for $y_1$ it follows that $cos\psi = cos\varphi \; cos \theta$...?
Edit: So the actual question is, why is $cos\psi = cos\varphi \; cos \theta$? Theta $(\theta)$ is the angle between $v_3$ and $y$, phi $(\varphi)$ is the angle between $y$ and $w_1$. Psi $(\psi)$ would be the angle between $v_3$ and $w_1$.
Now I think of it, I could've saved myself and you as reader some time since the actual problem only has to do with angles. Like I said, I know two angles $\varphi$ and $\theta$, how can I get $\psi$ from them such that the two expressions for $y_1$ are the same (and a similar question for $y_2$)?