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To determine the point \( p \) on line \( L \) that is closest to the point \((-16, 19, -17)\), we can approach the problem as follows:

### Step-by-Step Solution

1. **Define the Direction Vector of the Line \( L \):**
   The direction vector of line \( L \), which passes through the points \((12, 20, -14)\) and \((-9, 13, 11)\), is found by subtracting these points:
   \[
   \vec{d} = (-9 - 12, 13 - 20, 11 + 14) = (-21, -7, 25)
   \]

2. **Parametrize the Line \( L \):**
   A point \( (x, y, z) \) on \( L \) can be represented parametrically as:
   \[
   (x, y, z) = (12, 20, -14) + t(-21, -7, 25)
   \]
   So, the parametric equations of \( L \) are:
   \[
   x = 12 - 21t, \quad y = 20 - 7t, \quad z = -14 + 25t
   \]

3. **Find the Point on \( L \) Closest to \((-16, 19, -17)\):**
   The point on \( L \) closest to \((-16, 19, -17)\) is the projection of \((-16, 19, -17)\) onto the line.

4. **Set Up the Projection Formula:**
   Let \( \vec{a} = (12, 20, -14) \) (a point on \( L \)) and \( \vec{b} = (-16, 19, -17) \) (the point we're projecting). We want to find \( t \) such that the vector from \( \vec{a} \) to the projection point is parallel to \( \vec{d} \):
   \[
   t = \frac{(\vec{b} - \vec{a}) \cdot \vec{d}}{\vec{d} \cdot \vec{d}}
   \]
   where:
   \[
   \vec{b} - \vec{a} = (-16 - 12, 19 - 20, -17 + 14) = (-28, -1, -3)
   \]
   and
   \[
   \vec{d} \cdot \vec{d} = (-21)^2 + (-7)^2 + (25)^2 = 441 + 49 + 625 = 1115
   \]
   Now compute \( (\vec{b} - \vec{a}) \cdot \vec{d} \):
   \[
   (\vec{b} - \vec{a}) \cdot \vec{d} = (-28)(-21) + (-1)(-7) + (-3)(25) = 588 + 7 - 75 = 520
   \]
   Therefore,
   \[
   t = \frac{520}{1115}
   \]

5. **Calculate the Coordinates of \( p \):**
   Substitute \( t = \frac{520}{1115} \) into the parametric equations for \( x \), \( y \), and \( z \):
   \[
   p_1 = 12 - 21 \cdot \frac{520}{1115}
   \]
   \[
   p_2 = 20 - 7 \cdot \frac{520}{1115}
   \]
   \[
   p_3 = -14 + 25 \cdot \frac{520}{1115}
   \]

   Calculating each coordinate:

   - \( p_1 = 12 - 21 \cdot \frac{520}{1115} \approx 2.208 \)
   - \( p_2 = 20 - 7 \cdot \frac{520}{1115} \approx 16.74 \)
   - \( p_3 = -14 + 25 \cdot \frac{520}{1115} \approx -2.348 \)

### Final Answer
The coordinates of the point \( p \) are approximately:
\[
p = (2.208, 16.74, -2.348)
\]

Would you like more details on any step or have questions about this approach?

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1. What is the process of finding the direction vector for a line given two points?
2. How can parametric equations help in finding points on a line?
3. What is the purpose of the projection formula in vector calculations?
4. Why do we need to find the closest point on a line to another point in space?
5. How does the dot product assist in finding projections?

**Tip:** The projection of a point onto a line minimizes the distance between the point and any point on the line, making it useful for finding the closest point in three-dimensional space.

Take L to be the line that passes through (12, 20, -14) and (-9, 13, 11), and p to be the point on L that is closest to (-16, 19, -17). Determine the coordinates of p.

Discover how to find the coordinates of the point on a line closest to a given point in 3D space. This solution covers vector projection, parametric equations, and uses the projection formula to determine the exact coordinates.

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