I'm studying the bra-ket notation and I got a little confused by the assertion, that the scalar product $\left< x \middle| \psi \right>$ is a projection of the $\psi$ vector onto $x$. This is mentioned, for example, in this question.
But, the projection of vector $a$ onto a vector $b$ is usually defined as
$$ proj_b a = \frac{a \cdot b}{||b||}\frac{b}{||b||}. $$
So, how is it possible, that just the scalar product itself is considered a projection here?
On
If the length of the vector $b$ is $1$, then the projection is simply $$(a\cdot b)\cdot b$$
and the length of this vector is simply $a\cdot b$. So I suspect that $x$ in your case is normalized.
On
From the link that you shared, "In quantum mechanics the expression $\langle \phi | \psi \rangle$ is typically interpreted as the probability amplitude for the state $\psi$ to collapse into the state $\phi$. Mathematically, this means the coefficient for the projection of $\psi$ onto $\phi$. It is also described as the projection of state $\psi$ onto state $\phi$."
Also, note that quantum states have $2$-norm that are $1$.
If $||x||=1$ then the scalar product provides the length of the projection as you defined it, otherwise you just have to interpret it as the length of the projection that is scaled. This usually makes sense when the length of the two vectors is fixed, but their angle may change.