To find the projection of the line segments joining the given points onto the specified lines, we will follow a systematic approach. Let's break down the solution into two parts as per the question.
### Part (I)
**Step 1: Identify the points.**
Let the points be:
- \( A(2, -3, 0) \)
- \( B(0, 4, 5) \)
**Step 2: Find the direction vector of the line segment AB.**
The direction vector \( \overrightarrow{AB} \) can be calculated as:
\[
\overrightarrow{AB} = B - A = (0 - 2, 4 - (-3), 5 - 0) = (-2, 7, 5)
\]
**Step 3: Identify the direction cosines of the given line.**
The direction cosines are given as:
\[
\mathbf{l} = \left( \frac{2}{7}, \frac{3}{7}, -\frac{6}{7} \right)
\]
To find the direction vector \( \overrightarrow{B} \), we can multiply the direction cosines by a constant \( k \):
\[
\overrightarrow{B} = k \left( \frac{2}{7}, \frac{3}{7}, -\frac{6}{7} \right)
\]
For simplicity, we can take \( k = 7 \):
\[
\overrightarrow{B} = (2, 3, -6)
\]
**Step 4: Calculate the projection of \( \overrightarrow{AB} \) onto \( \overrightarrow{B} \).**
The formula for the projection of vector \( \overrightarrow{A} \) onto vector \( \overrightarrow{B} \) is given by:
\[
\text{proj}_{\overrightarrow{B}} \overrightarrow{A} = \frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|\overrightarrow{B}|^2} \overrightarrow{B}
\]
First, we need to calculate the dot product \( \overrightarrow{AB} \cdot \overrightarrow{B} \):
\[
\overrightarrow{AB} \cdot \overrightarrow{B} = (-2)(2) + (7)(3) + (5)(-6) = -4 + 21 - 30 = -13
\]
Next, we calculate the magnitude squared of \( \overrightarrow{B} \):
\[
|\overrightarrow{B}|^2 = 2^2 + 3^2 + (-6)^2 = 4 + 9 + 36 = 49
\]
Now, we can find the projection:
\[
\text{proj}_{\overrightarrow{B}} \overrightarrow{AB} = \frac{-13}{49} \overrightarrow{B}
\]
Thus, the projection is:
\[
\text{proj}_{\overrightarrow{B}} \overrightarrow{AB} = \frac{-13}{49} (2, 3, -6)
\]
### Part (II)
**Step 1: Identify the points.**
Let the points be:
- \( C(1, 2, 3) \)
- \( D(4, 3, 1) \)
**Step 2: Find the direction vector of the line segment CD.**
The direction vector \( \overrightarrow{CD} \) can be calculated as:
\[
\overrightarrow{CD} = D - C = (4 - 1, 3 - 2, 1 - 3) = (3, 1, -2)
\]
**Step 3: Identify the direction ratios of the given line.**
The direction ratios are given as:
\[
\mathbf{r} = (3, -6, 2)
\]
This can be treated as the direction vector \( \overrightarrow{R} \).
**Step 4: Calculate the projection of \( \overrightarrow{CD} \) onto \( \overrightarrow{R} \).**
Using the same projection formula:
First, we calculate the dot product \( \overrightarrow{CD} \cdot \overrightarrow{R} \):
\[
\overrightarrow{CD} \cdot \overrightarrow{R} = (3)(3) + (1)(-6) + (-2)(2) = 9 - 6 - 4 = -1
\]
Next, we calculate the magnitude squared of \( \overrightarrow{R} \):
\[
|\overrightarrow{R}|^2 = 3^2 + (-6)^2 + 2^2 = 9 + 36 + 4 = 49
\]
Now, we can find the projection:
\[
\text{proj}_{\overrightarrow{R}} \overrightarrow{CD} = \frac{-1}{49} \overrightarrow{R}
\]
Thus, the projection is:
\[
\text{proj}_{\overrightarrow{R}} \overrightarrow{CD} = \frac{-1}{49} (3, -6, 2)
\]
### Final Answers:
1. The projection of the line segment joining points \( (2, -3, 0) \) and \( (0, 4, 5) \) on the line with direction cosines \( \left( \frac{2}{7}, \frac{3}{7}, -\frac{6}{7} \right) \) is \( \frac{-13}{49} (2, 3, -6) \).
2. The projection of the line segment joining points \( (1, 2, 3) \) and \( (4, 3, 1) \) on the line with direction ratios \( (3, -6, 2) \) is \( \frac{-1}{49} (3, -6, 2) \).
