Question

For two vectors, what angle between them will lead to smallest magnitudes when added?

Answer 1

Let's consider the sum of two vectors $\mathbf{a}$ and $\mathbf{b}$ with the angle $\alpha$ between them:

Here $\mathbf{c} = \mathbf{a} +\mathbf{b}$. Using the law of cosines, we can find the length of the $\mathbf{c}$:

$$c^2 = a^2 +b^2-2ab\cos(180\degree-\alpha)$$

Using the trig identity:

$$\cos(180\degree-\alpha) = -\cos\alpha$$

obtain:

$$c^2 = a^2 +b^2+2ab\cos\alpha$$

In order to minimize the $c$, the $\alpha$ should be chosen equal to $180\degree$. Then $\cos\alpha = -1$ and

$$c^2 = a^2 +b^2-2ab$$

Answer. The angle 180 between two vectors will lead to smallest magnitudes when added.