Vectors `2hati+2hatj-2hatk,5hati+yhatj+hatk and -hati+2hatj+2hatk` are coplanar then find the value of y.

Explanation

To determine the value of \( y \) such that the vectors \( \mathbf{A} = 2\hat{i} + 2\hat{j} - 2\hat{k} \), \( \mathbf{B} = 5\hat{i} + y\hat{j} + \hat{k} \), and \( \mathbf{C} = -\hat{i} + 2\hat{j} + 2\hat{k} \) are coplanar, we can use the property that the scalar triple product of the vectors must be zero. This can be determined using the determinant of a matrix formed by the coefficients of the vectors.

### Step-by-Step Solution:

1. **Write the vectors in component form:**
\[ \mathbf{A} = (2, 2, -2), \quad \mathbf{B} = (5, y, 1), \quad \mathbf{C} = (-1, 2, 2) \]
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