Angle for n-fold Magnitude Increase of A→ +B→

[beauty-eyes]

Two vectors A→ and B→ have precisely equal magnitude .For the magnitude of A→ +B→ to be larger than the magnitude of
A→ - B→ by the factor n, what must be the angle between them?

 

[Dick]

Clearly, 1) the actual length of the vectors doesn't matter, so let's just make them length 1 and 2) since there are only two vectors involved, we can work in two dimensions. So call one vector (1,0) and the other one (cos(theta),sin(theta)). Now just set up your question and solve for theta.

 

[andrevdh]

The resultant of both constructions are diagonals of a rhombi.

This means that for the sum the resultant will be

$$2a\cos(\frac{\theta}{2})$$

where $$\theta$$ is the require angle and $$a$$ is the magnitude.

For the difference the angle between the vectors will be $$180^o - \theta$$. In this case the resultant amounts to

$$2a\sin(\frac{\theta}{2})$$