How Long Can The Product Go!!!

Geometry Level pending

Suppose a \(3\text{D}\) vector \(\vec{v}=(a,b,c)\) with \(a,b,c\geq 0\) makes angles \(\alpha,\beta,\gamma\) with the \(x,y,z\) axes respectively. The direction cosines of \(\vec{v}\) is then defined by the followings: \[\cos \alpha=\dfrac{a}{\sqrt{a^2+b^2+c^2}}\] \[\cos\beta=\dfrac{b}{\sqrt{a^2+b^2+c^2}}\] \[\cos \gamma=\dfrac{c}{\sqrt{a^2+b^2+c^2}}\] The maximum value of \(\cos\alpha \cos\beta\cos \gamma\) can be expressed as \(\dfrac{\sqrt{m}}{n}\) where \((m,n)\in\mathbb{N}^2\). Find the value of \(m+n\).

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