Let the coordinates of the points A, B, C be (1,8,4), (0,-11,4) and (2,-3,1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC ?

Options

A. (3,4,-2)
B. (4,-2,5)
C. (4,5,-2)
D. (2,4,5)

Explanation

To find the coordinates of point D, which is the foot of the perpendicular from point A to line segment BC, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A, B, and C:**
- A(1, 8, 4)
- B(0, -11, 4)
- C(2, -3, 1)

2. **Find the Direction Vector of Line BC:**
- The direction vector \( \vec{BC} \) can be calculated as:
\[ \vec{BC} = C - B = (2 - 0, -3 - (-11), 1 - 4) = (2, 8, -3) \]

3. **Set Up the Equation for the Perpendicular Line AD:**
- Let the coordinates of point D be \( D(\alpha, \beta, \gamma) \).
- The direction vector \( \vec{AD} \) is given by:
\[ \vec{AD} = D - A = (\alpha - 1, \beta - 8, \gamma - 4) \]

4. **Use the Condition for Perpendicular Vectors:**
- For vectors \( \vec{AD} \) and \( \vec{BC} \) to be perpendicular, their dot product must equal zero:
\[ \vec{AD} \cdot \vec{BC} = 0 \]
- This gives us the equation:
\[ (\alpha - 1) \cdot 2 + (\beta - 8) \cdot 8 + (\gamma - 4) \cdot (-3) = 0 \]

5. **Expand the Equation:**
- Expanding the equation, we get:
\[ 2(\alpha - 1) + 8(\beta - 8) - 3(\gamma - 4) = 0 \]
- Simplifying this, we have:
\[ 2\alpha - 2 + 8\beta - 64 - 3\gamma + 12 = 0 \]
- Rearranging gives:
\[ 2\alpha + 8\beta - 3\gamma - 54 = 0 \]

6. **Substituting Possible Coordinates for D:**
- We will check the given options to find the coordinates that satisfy the equation \( 2\alpha + 8\beta - 3\gamma - 54 = 0 \).

- **Option 1: D(3, 4, -2)**
\[ 2(3) + 8(4) - 3(-2) - 54 = 6 + 32 + 6 - 54 = 0 \quad \text{(Valid)} \]

- **Option 2: D(4, -2, 5)**
\[ 2(4) + 8(-2) - 3(5) - 54 = 8 - 16 - 15 - 54 \neq 0 \quad \text{(Invalid)} \]

- **Option 3: D(4, 5, -2)**
\[ 2(4) + 8(5) - 3(-2) - 54 = 8 + 40 + 6 - 54 \neq 0 \quad \text{(Invalid)} \]

7. **Conclusion:**
- The coordinates of point D, which is the foot of the perpendicular from A to line BC, are \( D(3, 4, -2) \).