Question

if a+b+c=0, |a|=3, |b|=5 and |c|=7, then find the angle between a and b.

Answer 1

if $a + b + c = 0$, $|a| = 3$, $|b| = 5$ and $|c| = 7$, then find the angle between $a$ and $b$.

$$c^2 = a^2 + b^2 - 2ab * cos\gamma$$ $$7^2 = 3^2 + 5^2 - 2 * 3 * 5 * cos\gamma$$ $$49 = 9 + 25 - 30 * cos\gamma$$ $$49 - 34 = -30 * cos\gamma$$ $$15 = -30 * cos\gamma$$ $$-\frac{1}{2} = cos\gamma$$ $$\gamma = \arccos\left(-\frac{1}{2}\right)$$ $$\gamma = \frac{2\pi}{3} = 120{}^\circ$$

Answer:

$$\gamma = 120{}^\circ$$