Vector a = (2, 1, 2) Vector b = (1, 2, 4) Vector p = (k, k, k)
Vector a to vector b = vector b - vector a = (1, 2, 4) - (2, 1, 2) = (1 - 2, 2 - 1, 4 - 2) = (-1, 1, 2) Vector a to vector p = vector p - vector a = (k, k, k) - (2, 1, 2) = (k - 2, k - 1, k - 2)
Vector a to b is perpendicular to vector a to p if the dot product of vector a to vector b and vector a to vector p is equal to zero. i.e. (-1, 1, 2) . (k - 2, k - 1, k - 2) = 0 -1(k - 2) + (k - 1) + 2(k - 2) = 0 -k + 2 + k - 1 + 2k - 4 = 0 2k -3 = 0 2k = 3 k = 3/2
First two rolls have to be 1-4 that is 2/3 chance twice and the third can be 4or 5 2/3*2/3*1/3 + the chance that the fourth is the 5 or 6. 2/3*2/3*2/3*1/3
So the solution is : P=2/3*2/3*1/3 + 2/3*2/3*2/3*1/3