Find the value of 'x' for which the four points : `A(x, -1, -1), B(4, 5, 1), C(3, 9, 4)` and `D(-4, 4, 4)` are coplanar.

Answer

To find the value of 'x' for which the points A(x, -1, -1), B(4, 5, 1), C(3, 9, 4), and D(-4, 4, 4) are coplanar, we will use the concept of the scalar triple product. The points are coplanar if the scalar triple product of the vectors AB, AC, and AD is equal to zero.

### Step-by-Step Solution:

1. **Find the position vectors of the points:**
- \( A = (x, -1, -1) \)
- \( B = (4, 5, 1) \)
- \( C = (3, 9, 4) \)
- \( D = (-4, 4, 4) \)

2. **Calculate the vectors AB, AC, and AD:**
- **Vector AB**:
\[ \vec{AB} = \vec{B} - \vec{A} = (4 - x, 5 - (-1), 1 - (-1)) = (4 - x, 6, 2) \]
- **Vector AC**:
\[ \vec{AC} = \vec{C} - \vec{A} = (3 - x, 9 - (-1), 4 - (-1)) = (3 - x, 10, 5) \]
- **Vector AD**:
\[ \vec{AD} = \vec{D} - \vec{A} = (-4 - x, 4 - (-1), 4 - (-1)) = (-4 - x, 5, 5) \]

3. **Set up the scalar triple product (determinant):**
The scalar triple product can be represented as the determinant of the matrix formed by the vectors AB, AC, and AD:
\[ \begin{vmatrix} 4 - x & 6 & 2 \\ 3 - x & 10 & 5 \\ -4 - x & 5 & 5 \end{vmatrix} = 0 \]

4. **Calculate the determinant:**
Expanding the determinant:
\[ = (4 - x) \begin{vmatrix} 10 & 5 \\ 5 & 5 \end{vmatrix} - 6 \begin{vmatrix} 3 - x & 5 \\ -4 - x & 5 \end{vmatrix} + 2 \begin{vmatrix} 3 - x & 10 \\ -4 - x & 5 \end{vmatrix} \]

Calculate each of the 2x2 determinants:
- \( \begin{vmatrix} 10 & 5 \\ 5 & 5 \end{vmatrix} = (10)(5) - (5)(5) = 50 - 25 = 25 \)

- \( \begin{vmatrix} 3 - x & 5 \\ -4 - x & 5 \end{vmatrix} = (3 - x)(5) - (5)(-4 - x) = 15 - 5x + 20 + 5x = 35 \)

- \( \begin{vmatrix} 3 - x & 10 \\ -4 - x & 5 \end{vmatrix} = (3 - x)(5) - (10)(-4 - x) = 15 - 5x + 40 + 10x = 55 + 5x \)

Substitute back into the determinant:
\[ (4 - x)(25) - 6(35) + 2(55 + 5x) = 0 \]
\[ 25(4 - x) - 210 + 110 + 10x = 0 \]
\[ 100 - 25x - 210 + 110 + 10x = 0 \]
Combine like terms:
\[ -15x + 0 = 0 \]
Thus,
\[ -15x = 0 \implies x = 0 \]

5. **Conclusion:**
The value of \( x \) for which the points A, B, C, and D are coplanar is \( x = 0 \).