If points A (2, 4, 0), B(3, 1, 8), C(3, 1, -3), D(7, -3, 4) are four points then projection of line segment AB on line CD.

Explanation

To find the projection of the line segment AB on the line segment CD, we can follow these steps:

### Step 1: Find the vector AB
The vector AB can be calculated using the coordinates of points A and B. The formula for finding the vector from point A(x1, y1, z1) to point B(x2, y2, z2) is:

\[ \vec{AB} = (x2 - x1) \hat{i} + (y2 - y1) \hat{j} + (z2 - z1) \hat{k} \]

Substituting the coordinates of points A(2, 4, 0) and B(3, 1, 8):

\[ \vec{AB} = (3 - 2) \hat{i} + (1 - 4) \hat{j} + (8 - 0) \hat{k} \]
\[ \vec{AB} = 1 \hat{i} - 3 \hat{j} + 8 \hat{k} \]

### Step 2: Find the vector CD
Similarly, we find the vector CD using the coordinates of points C and D. The formula is the same:

\[ \vec{CD} = (x4 - x3) \hat{i} + (y4 - y3) \hat{j} + (z4 - z3) \hat{k} \]

Substituting the coordinates of points C(3, 1, -3) and D(7, -3, 4):

\[ \vec{CD} = (7 - 3) \hat{i} + (-3 - 1) \hat{j} + (4 - (-3)) \hat{k} \]
\[ \vec{CD} = 4 \hat{i} - 4 \hat{j} + 7 \hat{k} \]

### Step 3: Calculate the projection of vector AB on vector CD
The projection of vector AB onto vector CD is given by the formula:

\[ \text{Projection of } \vec{AB} \text{ on } \vec{CD} = \frac{\vec{AB} \cdot \vec{CD}}{|\vec{CD}|} \]

First, we need to calculate the dot product \(\vec{AB} \cdot \vec{CD}\):

\[ \vec{AB} \cdot \vec{CD} = (1)(4) + (-3)(-4) + (8)(7) \]
\[ = 4 + 12 + 56 = 72 \]

Next, we calculate the magnitude of vector CD:

\[ |\vec{CD}| = \sqrt{(4^2) + (-4^2) + (7^2)} = \sqrt{16 + 16 + 49} = \sqrt{81} = 9 \]

Now we can find the projection:

\[ \text{Projection of } \vec{AB} \text{ on } \vec{CD} = \frac{72}{9} = 8 \]

### Final Answer:
The projection of the line segment AB on the line segment CD is **8**.

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