Maximizing Vector Sum with Constant Magnitudes

[ForceBoy]

<Moderator's note: Moved from a technical forum and thus no template.>

Let there be two vectors, u and v. Whose magnitudes are constant

u = [a, b]
v = [x, y]

Define c = ||u|| and k = ||v||

Now sum the vectors:

w = u + v = [a, b] +[x, y] = [a+x, b+y]

Now find ||w||

||w|| =√(a+x)²+(b+y)²

||w|| = √a²+2ax+x²+b²+2by+y²

||w|| = √u⋅u + v⋅v +2(ax+by)

||w|| = √u⋅u + v⋅v +2(u⋅v)

||w|| = √u⋅u + v⋅v +2||u|| ||v|| cos Θ

||w|| = √u⋅u + v⋅v +2ck cos Θ

Here is where I have trouble. I want to find the angle between the two vectors that would make ||w|| take on the largest possible value. I know that to do so I have to differentiate with respect to Θ. What I am not sure about is whether I would treat u and v as constants. Also I apologize if this does not belong in this forum. I was torn between this forum and the calculus forum.

EDIT:
Disregard that last part. This post was moved from the Linear algebra thread

 

[fresh_42]

I don't think you need the derivative of $\cos \Theta$. Just ask yourself, when does the cosine have its maximal values?

 

[ForceBoy]

Thank you. That's much simpler. I was overthinking it.

 

[FactChecker]

Even then, you asked if you can consider the vectors to be constant or not. That is still an important issue no matter what approach you use. You can not consider them to be constant, since you are considering different angles between them. But you can consider their norms to be constant regardless of which way they are pointing. That is all you need. To be picky, you should put the equation in a form that does not depend on the exact vectors but just on their norms before you state that the norms are constant.