In tetrahedron OABC with O as the origin, A = (1,0,0), B = (1,1,0), C = (0,1,1), find the distance lambda of O from the plane ABC. Then determine the nature of 1/lambda².

Prime number
Even number
Odd number
Composite number

Correct answer: Prime number

Solution

Normal n = AB x AC = (1,0,1). Plane: 1(x-1)+0(y-0)+1(z-0)=0 -> x+z=1. Distance from O(0,0,0): |0+0-1|/sqrt(2) = 1/sqrt(2). So lambda = 1/sqrt(2), lambda² = 1/2, 1/lambda² = 2 which is a prime number.

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