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Simon Bridge
Homework Helper
when it comes to this particular question
Those are three different, but related, questions about vector manipulation. Not one particular question.

Q1. Is there any systematic manner on how to prove [geometric theorems] using vectors?
A. No.

Q2. Find y component of vector C from its length and the angle it makes with the x axis
$$C_y = C\sin\theta$$ ... provided in the thread (post #6).
You should be able to figure this out from the trig relations - remember SOH CAH TOA?

Q3. this last is for vector addition using geometry.
You do this numerically by resolving the vectors into components and using normal addition.
Also be scale diagram, drawing the vectors head-to-tail, and measuring.

In this specific example C=A+B where angle between them is θ - put one vector along the x axis and resolve components of the other one.$$C = \sqrt{(A+B\cos\theta)^2 + (B\sin\theta)^2}=\sqrt{A^2 + B^2 + 2AB\cos\theta}$$$$\phi = \tan^{-1}\bigg ( \frac{B\sin\theta}{A+B\cos\theta} \bigg )$$so vector C will have magnitude $C$ and will have an angle $\phi$ to A.
The lynchpin is given in post #3.

How come ... there are many threads by various people, but never an answer or explanation in response to this question?
Just going by your examples, this is probably because we don't do your homework for you. The only way you'll get the kind of result you are looking for is if the OP posts it.
Q1. not specific enough to answer - OP was asked for examples and chose not to reply.
Q2 and 3. both homework: against the rules.

Too many people just abandon a thread when they have enough information to solve their problem instead of posting the solution they came up with.

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