Calculate the value of a for the points (1, 0, 1), (0, 1, 2), (-1,2,a) and (7, 2, 1) so that they are 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 three vectors A, B and C is given by A.(BxC).

So, we calculate BxC first:

BxC = (20 - (a-1)2, -20 - 6(a-1), -2*6 - (-2)*2) = (-2a+2, -6a+6, -12)

Then, we calculate A.(BxC):

A.(BxC)