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# Long hard answer pls Very hard Vector

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A vector v is called a unit vector if $$\|{v}\| = 1$$.

Let a,b , and c be unit vectors, such that a+b+c=0, Show that the angle between any two of these vectors is $$120^\circ$$.

(try to use vectors!)

Feb 20, 2019

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$$a + b + c = 0\\ a + b = -c\\ (a+b)\cdot (a+b) = (-c)\cdot (-c)\\ \|a\|^2 + 2 a\cdot b + \|b\|^2 = \|c\|^2\\ 1 + 2a \cdot b + 1 = 1\\ 2a\cdot b = -1\\ a \cdot b = -\dfrac 1 2$$

$$\text{the angle between }a \text{ and }b \text{ is given by}\\ \cos(\theta) = \dfrac{a\cdot b}{\|a\|\|b\|} = \dfrac{-\frac 1 2}{1} = -\dfrac 1 2\\ \theta = \arccos\left(-\dfrac 1 2\right) = \dfrac{2\pi}{3} = 120^\circ$$

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Feb 20, 2019