Proving $|A\cdot B|\leq \|A\|\|B\|$

Question

I have been reading Introduction to Linear Algebra by Serge Lang and cannot follow his proof of this property of norms of vectors. In particular, he introduces $c$ as a component of $A$ and $B$ and then define $c$ as if it is a projection of the vector. Why is this the case?

Answer 1

In the proof, the vector $\mathbf A$ can be thought of being composed of two components---one parallel to $\mathbf B$ and one perpendicular to $\mathbf B$, thereby forming a right-angled triangle. The vector sum of these two components is equal to the original vector $\mathbf A$.

Now that we have a right-angled triangle, we can apply Pythogoras theorem.