A line makes angles \alpha, \beta and \gamma with the positive directions of the coordinate axes. If \vec{a}=(\sin^{2}\alpha)\hat{i}+(\sin^{2}\beta)\hat{j}+(\sin^{2}\gamma)\hat{k} and \vec{b}=\hat{i}+\hat{j}+\hat{k}, then what is \vec{a}\cdot\vec{b} equal to?

Explanation

The dot product \vec{a}\cdot\vec{b} = \sin^2\alpha + \sin^2\beta + \sin^2\gamma. Using the identity \sin^2\theta = 1 - \cos^2\theta, this becomes 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma). For a line in 3D space, the sum of squares of direction cosines is 1. Thus, \vec{a}\cdot\vec{b} = 3 - 1 = 2.

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