What is the angle between two vector components?

[archa1c]

Homework Statement

A velocity of $$10ms^{-1}$$ is to be replaced by two components, $$7.0ms^{-1}$$ and $$5.0ms^{-1}$$. What must be the angle between the two components?

Homework Equations

The Attempt at a Solution

Now I think that the answer to the solution lies in using trig to work out the angles, and that solving this equation $$(5 sin\Theta)^2 + (7 + 5cos\Theta)^2 = 10^2$$ should give me the respective answers. What I don't understand is WHY I am doing that. So if someone could be so kind as to tell me how I would reach the conclusion that I should do those steps I would be very grateful.

 

[physicsnoob93]

Hmm, you should use the fact that

Vx=Vcos(theta)
Vy=Vsin(theta) and that Vx^2+Vy^2 = V^2
where Theta is the angle between the components. I'm not sure why you have a (7+5cos(\theta))^2 there.

 

[archa1c]

Well in my book I am basically told to use this equations (Where V_r is the resultant):

$$V_r^2 = (V_1 sin\Theta)^2 + (V_1 cos\Theta + V_2)^2$$ - I want to know why I would use this formula...

 

[physicsnoob93]

Have you learned something called the cosine rule before? The cosine rule says,
that if I have 2 vectors a and b, |a+b|^2=a^2+b^2+2abcos(theta).

Now, relate that to the formula written in the book.