Calculate the value of a for the points (1, 0, 1), (0, 1, 2), (-1, 2, a) and (7, 2, 1) so that theyare coplanar.

To find the value of 'a' for the points to be coplanar, we can use the scalar triple product method. The points are coplanar if the scalar triple product of the vectors formed by these points is zero.

The vectors can be formed as follows:

Vector A = (0-1, 1-0, 2-1) = (-1, 1, 1) Vector B = (-1-1, 2-0, a-1) = (-2, 2, a-1) Vector C = (7-1, 2-0, 1-1) = (6, 2, 0)

The scalar triple product of A, B, and C is given by:

A.(BxC) = A. [(2, 2, a-1) x (6, 2, 0)] = A. [(20 - (a-1)2, (a-1)6 - 20, 26 - 22)] = A. [(-2a+2, 6a-6, 8)] = (-1, 1, 1).(-2a+2, 6a-6, 8) = -1*(-2a+2) + 1*(6a-6) + 1*8 = 2a - 2 + 6a - 6 + 8 = 8a - 0

For the points to be coplanar, A.(BxC) should be equal to 0.

So, 8a - 0 = 0 => 8a = 0 => a = 0/8 => a = 0

So, the value of 'a' for the points to be coplanar is 0.