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

Options

A. The compound of `vec(a)-vec(b)` will be `(a^(2)-b^(2))/sqrt(a^(2)+b^(2)+ab)`
B. `vec(a)xx vec(b)` is perpendicular to resultant of `(vec(a)+2vec(b))` and `(vec(a)- vec(b))`
C. The component of `vec(a)-vec(b)` along `vec(a)+vec(b)` will be `(a^(2)-b^(2))/sqrt(a^(2)+b^(2)+2ab)`
D. The component of `vec(a)+vec(b)` along `vec(a)-vec(b)` will be `(a^(2)-b^(2))/sqrt(a^(2)+b^(2)+sqrt(3)ab)`

Explanation

To determine the value of \( y \) for which the vectors \( \mathbf{A} = 5\hat{i} + y\hat{j} + \hat{k} \), \( \mathbf{B} = 2\hat{i} + 2\hat{j} - 2\hat{k} \), and \( \mathbf{C} = -\hat{i} + 2\hat{j} + 2\hat{k} \) are coplanar, we can use the property that three vectors are coplanar if the scalar triple product of the vectors is zero. This can be represented using a determinant.

### Step-by-Step Solution:

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

2. **Set up the determinant**:
The vectors are coplanar if the determinant of the matrix formed by their components is zero:
\[ \begin{vmatrix} 5 & y & 1 \\ 2 & 2 & -2 \\ -1 & 2 & 2 \end{vmatrix} = 0 \]

3. **Calculate the determinant**:
Expanding the determinant:
\[ = 5 \begin{vmatrix} 2 & -2 \\ 2 & 2 \end{vmatrix} - y \begin{vmatrix} 2 & -2 \\ -1 & 2 \end{vmatrix} + 1 \begin{vmatrix} 2 & 2 \\ -1 & 2 \end{vmatrix} \]

Now, calculate each of the 2x2 determinants:
- \( \begin{vmatrix} 2 & -2 \\ 2 & 2 \end{vmatrix} = (2)(2) - (-2)(2) = 4 + 4 = 8 \)
- \( \begin{vmatrix} 2 & -2 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-2)(-1) = 4 - 2 = 2 \)
- \( \begin{vmatrix} 2 & 2 \\ -1 & 2 \end{vmatrix} = (2)(2) - (2)(-1) = 4 + 2 = 6 \)

Substitute these values back into the determinant:
\[ = 5(8) - y(2) + 1(6) = 40 - 2y + 6 \]
\[ = 46 - 2y \]

4. **Set the determinant equal to zero**:
\[ 46 - 2y = 0 \]

5. **Solve for \( y \)**:
\[ 2y = 46 \implies y = \frac{46}{2} = 23 \]

### Final Answer:
The value of \( y \) for which the vectors are coplanar is \( y = 23 \).