A(-2,3,4), B(1,1,2),C(4,-1,0) are three points. Find the Cartesian equations of line AB and show that points A,B,C are collinear.

Answer

To solve the problem, we need to find the Cartesian equations of the line AB and then show that points A, B, and C are collinear.

### Step 1: Identify the coordinates of points A and B
We have the coordinates of points A and B as follows:
- A(-2, 3, 4)
- B(1, 1, 2)

### Step 2: Use the formula for the Cartesian equation of the line
The Cartesian equation of a line passing through two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by:
\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \]

### Step 3: Substitute the coordinates of points A and B into the formula
Here, we can substitute:
- \( x_1 = -2, y_1 = 3, z_1 = 4 \)
- \( x_2 = 1, y_2 = 1, z_2 = 2 \)

So, we have:
\[ \frac{x - (-2)}{1 - (-2)} = \frac{y - 3}{1 - 3} = \frac{z - 4}{2 - 4} \]

### Step 4: Simplify the equations
This simplifies to:
\[ \frac{x + 2}{3} = \frac{y - 3}{-2} = \frac{z - 4}{-2} \]

### Step 5: Write the Cartesian equations
Thus, the Cartesian equations of line AB can be written as:
\[ \frac{x + 2}{3} = \frac{y - 3}{-2} = \frac{z - 4}{-2} \]

### Step 6: Show that points A, B, and C are collinear
Now we need to check if point C(4, -1, 0) lies on the line AB.

Substituting the coordinates of point C into the equations:
1. For \( x \):
\[ \frac{4 + 2}{3} = \frac{6}{3} = 2 \]
2. For \( y \):
\[ \frac{-1 - 3}{-2} = \frac{-4}{-2} = 2 \]
3. For \( z \):
\[ \frac{0 - 4}{-2} = \frac{-4}{-2} = 2 \]

### Step 7: Conclusion
Since all three equations yield the same value (2), point C satisfies the equation of line AB. Therefore, points A, B, and C are collinear.

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