# Projection on a Plane

Geometry Level 5

Suppose, $$\hat{a}, \hat{b}$$ and $$\hat{c}$$ are three non-coplanar unit vectors. Let $$\theta_{ab}, \theta_{ac}$$ and $$\theta_{bc}$$ be the angles between $$\hat{a}$$ & $$\hat{b}$$, $$\hat{a}$$ & $$\hat{c}$$ and $$\hat{b}$$ & $$\hat{c}$$ respectively. Let $$P$$ be a plane on which $$\hat{b}$$ and $$\hat{c}$$ lie. The projection vector of $$\hat{a}$$ on the plane $$P$$ is, $\vec{A_p}=B\hat{b}+C\hat{c}$ Given that, $$\cos {\theta_{ab}} = \frac{1}{5}, cos{\theta_{ac}}= \frac{1}{6}$$ and $$\cos{\theta_{bc}}=\frac{1}{2}$$.

$$B-C$$ can be expressed as $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers. Calculate the value of $$m+n$$.

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