Orthogonal Vectors:
Two vectors are said to be orthogonal if their dot product is zero. In other words, the angle between them is 90 degrees.

Given Vectors:
Let's consider the given vectors:
A = 3i + 2j + 6k
B = 2i + j + k
C = i - j + k

Finding the Orthogonal Vector:
To find a vector that is orthogonal to the vector A and coplanar with vectors B and C, we can take the cross product of vectors B and C. The cross product of two vectors gives a vector that is perpendicular (orthogonal) to both of them.

Cross Product:
The cross product of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is given by:
A × B = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k

Calculating the Cross Product:
Let's calculate the cross product of vectors B and C:
B × C = (1)(1) - (-1)(1)i - (1)(1) - (1)(-1)j + (2)(-1) - (1)(1)k
= 2i + 0j - 3k
= 2i - 3k

Unit Vector:
To obtain the unit vector, we divide the vector by its magnitude. The magnitude of vector 2i - 3k is given by:
|2i - 3k| = √((2)^2 + (0)^2 + (-3)^2) = √(4 + 0 + 9) = √13

So, the unit vector that is orthogonal to the vector A and coplanar with vectors B and C is:
(2i - 3k) / √13

Summary:
The unit vector that is orthogonal to the vector 3i + 2j + 6k and coplanar with the vectors 2i + j + k and i - j + k is (2i - 3k) / √13.