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I need to solve the following problem. Which approach should I use ? Is there any simple procedure for such problem ?

Determine the angle between vectors $\vec{a}$ and $\vec{b}$ if:
$(\vec{a} + 3\vec{b}) \perp (7\vec{a} - 5\vec{b})$
and
$(\vec{a} - 4\vec{b}) \perp (7\vec{a} - 2\vec{b})$

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Perpendicular means that the dot product is 0. So expanding we get two equations relating a.b to a^2 and b^2. A little manipulation give that a^2=b^2 and a.b = (1/2) b^2. In other words, the vectors are the same length. But a.b = a b cos k, where k is the angle between them, so cos k = 0.5 and hence k = 60 degrees.

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  • $\begingroup$ great ! thanks very much. $\endgroup$ – user2778945 Sep 26 '15 at 19:32
  • $\begingroup$ @almagest I don't follow. Would you expand your answer to clarify how you get $a^2=b^2$? $\endgroup$ – Tim Thayer Sep 26 '15 at 21:21

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