P,Q,R,S are four vectors of equal magnitude. P + Q - R = 0, where the angle between P and Q is `theta_1.`  P + Q - S = 0, where the angle between P and S is `theta_2.` What is the ratio of `theta_1` and `theta_2?`

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First, rewrite the equalities as P + Q = R and P + Q = S. We see now that R = S, so `theta_1` is the angle between P and Q, `theta_2` is the angle between P and P + Q.

P + Q is the diagonal of...

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Hello!

First, rewrite the equalities as P + Q = R and P + Q = S. We see now that R = S, so `theta_1` is the angle between P and Q, `theta_2` is the angle between P and P + Q.

P + Q is the diagonal of the parallelogram formed by P and Q. Because P and Q have the equal magnitudes, this parallelogram is a rhombus, and its diagonals bisect the angles between the sides.

Thus `theta_2` is a half of `theta_1` , and the ratio is `theta_1` : `theta_2` is 2 : 1.

If we take into account that the magnitude of P+Q is the same of P and Q, we find that `theta_2=pi/3` and `theta_1=(2 pi)/3.`

 

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