Two forces $$\vec{F_1}$$ and $$\vec{F_2}$$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $$\vec{F_1}$$ and $$\vec{F_2}$$ is $$\cos^{-1}\left(\frac{1}{n}\right)$$. The value of |n| is _____.

Let $$F_1 = F$$ and $$F_2 = 3F$$. The resultant equals the larger force: $$R = 3F$$.

Using the formula: $$R^2 = F_1^2 + F_2^2 + 2F_1F_2\cos\theta$$

$$9F^2 = F^2 + 9F^2 + 6F^2\cos\theta$$

$$0 = F^2 + 6F^2\cos\theta$$

$$\cos\theta = -\frac{1}{6}$$

So $$\theta = \cos^{-1}\left(\frac{1}{n}\right)$$ where $$n = -6$$, and $$|n| = 6$$.

The answer is 6.