Let a=(? 2,1,2, b=(? 1,2,4, and p=(k, k,k. the vector from a to b is perpendicular to the vector from a to p when k =

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