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?

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.