If p is the perpendicular distance from origin to the plane passing through (1,0,0), (0,1,0) and (0,0,1), then what is 3p^{2} equal to?

Explanation

The equation of the plane with intercepts 1, 1, 1 is \frac{x}{1} + \frac{y}{1} + \frac{z}{1} = 1, which simplifies to x+y+z-1=0. The perpendicular distance p from the origin (0,0,0) to this plane is p = \frac{|0+0+0-1|}{\sqrt{1^2+1^2+1^2}} = \frac{1}{\sqrt{3}}. Squaring both sides yields p^2 = \frac{1}{3}, meaning 3p^2 = 1.

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