Points A(4, 5, 1), B (0, - 1, - 1), C (3 , 9, 4) and D( -4, 4, 4) are

Options

A. collinear
B. coplanar
C. non -coplanar
D. non-collinear and non-coplanar

Answer

To determine whether the points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4), and D(-4, 4, 4) are collinear, coplanar, or non-coplanar, we will follow these steps:

### Step 1: Find the position vectors of points A, B, C, and D
The position vectors can be represented as:
- **A** = \( \vec{A} = (4, 5, 1) \)
- **B** = \( \vec{B} = (0, -1, -1) \)
- **C** = \( \vec{C} = (3, 9, 4) \)
- **D** = \( \vec{D} = (-4, 4, 4) \)

### Step 2: Calculate the vectors AB, AC, and AD
To find the vectors, we use the formula \( \vec{AB} = \vec{B} - \vec{A} \), \( \vec{AC} = \vec{C} - \vec{A} \), and \( \vec{AD} = \vec{D} - \vec{A} \).

1. **Vector AB**:
\[ \vec{AB} = \vec{B} - \vec{A} = (0 - 4, -1 - 5, -1 - 1) = (-4, -6, -2) \]

2. **Vector AC**:
\[ \vec{AC} = \vec{C} - \vec{A} = (3 - 4, 9 - 5, 4 - 1) = (-1, 4, 3) \]

3. **Vector AD**:
\[ \vec{AD} = \vec{D} - \vec{A} = (-4 - 4, 4 - 5, 4 - 1) = (-8, -1, 3) \]

### Step 3: Form the matrix for the vectors
We will form a matrix using the vectors \( \vec{AB} \), \( \vec{AC} \), and \( \vec{AD} \):
\[ \begin{vmatrix} -4 & -6 & -2 \\ -1 & 4 & 3 \\ -8 & -1 & 3 \end{vmatrix} \]

### Step 4: Calculate the determinant
To check for coplanarity, we need to calculate the determinant of the matrix. If the determinant is zero, the points are coplanar.

Expanding the determinant:
\[ D = -4 \begin{vmatrix} 4 & 3 \\ -1 & 3 \end{vmatrix} + 6 \begin{vmatrix} -1 & 3 \\ -8 & 3 \end{vmatrix} - 2 \begin{vmatrix} -1 & 4 \\ -8 & -1 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 4 & 3 \\ -1 & 3 \end{vmatrix} = (4 \cdot 3) - (3 \cdot -1) = 12 + 3 = 15 \)

2. \( \begin{vmatrix} -1 & 3 \\ -8 & 3 \end{vmatrix} = (-1 \cdot 3) - (3 \cdot -8) = -3 + 24 = 21 \)

3. \( \begin{vmatrix} -1 & 4 \\ -8 & -1 \end{vmatrix} = (-1 \cdot -1) - (4 \cdot -8) = 1 + 32 = 33 \)

Now substituting back:
\[ D = -4(15) + 6(21) - 2(33) \]
\[ D = -60 + 126 - 66 \]
\[ D = 0 \]

### Conclusion
Since the determinant \( D = 0 \), the points A, B, C, and D are coplanar.

### Final Answer
The points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4), and D(-4, 4, 4) are **coplanar**.

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