Step 1: Understanding the Concept:
We need to find the length of the projection of a vector segment onto a given line.
First, we find the vector representing the line segment joining the two points. Then, we find the unit vector along the given line. The absolute value of the dot product of these two vectors gives the projection length.
Step 2: Key Formula or Approach:
Let the points be \( P \) and \( Q \). The vector \( \vec{PQ} = \vec{r}_Q - \vec{r}_P \).
Let the direction vector of the line be \( \vec{d} \).
The length of the projection of \( \vec{PQ} \) on the line is given by \( \frac{|\vec{PQ} \cdot \vec{d}|}{|\vec{d}|} \).
Step 3: Detailed Explanation:
Let the points be \( P(2, 1, -3) \) and \( Q(-1, 0, 2) \).
The vector \( \vec{PQ} \) is:
\[ \vec{PQ} = \langle -1 - 2, 0 - 1, 2 - (-3) \rangle = \langle -3, -1, 5 \rangle \]
\[ \vec{PQ} = -3\hat{i} - \hat{j} + 5\hat{k} \]
The line has direction ratios \( 3, 2, 6 \), so its direction vector is:
\[ \vec{d} = 3\hat{i} + 2\hat{j} + 6\hat{k} \]
The magnitude of vector \( \vec{d} \) is:
\[ |\vec{d}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]
Now, calculate the dot product \( \vec{PQ} \cdot \vec{d} \):
\[ \vec{PQ} \cdot \vec{d} = (-3)(3) + (-1)(2) + (5)(6) \]
\[ \vec{PQ} \cdot \vec{d} = -9 - 2 + 30 = 19 \]
The projection length is the absolute value of the dot product divided by the magnitude of the direction vector:
\[ \text{Projection} = \frac{|\vec{PQ} \cdot \vec{d}|}{|\vec{d}|} = \frac{|19|}{7} = \frac{19}{7} \]
Step 4: Final Answer:
The projection is \( \frac{19}{7} \) units.