What is the angle between two vector components?

AI Thread Summary
To find the angle between two vector components of a velocity of 10 m/s represented by 7.0 m/s and 5.0 m/s, the discussion emphasizes using trigonometric relationships. The equation (5 sin(Θ))^2 + (7 + 5cos(Θ))^2 = 10^2 is derived from the Pythagorean theorem and the cosine rule. Participants clarify the reasoning behind using this equation, linking it to the cosine rule which relates the magnitudes of the vectors and the angle between them. Understanding these relationships is crucial for solving the problem accurately. The discussion ultimately focuses on applying these mathematical principles to determine the required angle.
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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.
 
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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.
 
Well in my book I am basically told to use this equations (Where Vr 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 forumla...
 
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.
 
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