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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