Answer to Question #143798 in Mechanics | Relativity for Phoebe

Question

If the magnitude of the sum of two vectors is less than the magnitude of either vector, then:

Answer 1

The magnitude of the sum of two vectors can be found from the cosine law applied to $\triangle AOB$:

$$c = \sqrt{a^2 + b^2 - 2ab\cos(180-\theta)} = \sqrt{a^2 + b^2 + 2ab\cos\theta}$$

where $\theta$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$.

Thus, it should be $\theta >90\degree$ for $c<a$ and $c<b$.

Answer. If the magnitude of the sum of two vectors is less than the magnitude of either vector, then the angle between these two vectors is obtuse ($>90\degree$ ).