The problem asks for the equation of the line AB given points A(1, 2, 4) and B(-1, 5, 3).

1. Problem Description

The problem asks for the equation of the line AB given points A(1, 2, 4) and B(-1, 5, 3).

2. Solution Steps

First, we need to find the direction vector $\vec{AB}$ of the line AB.

$$\vec{AB} = B - A = (-1 - 1, 5 - 2, 3 - 4) = (-2, 3, -1)$$

Now, we can write the parametric equation of the line AB. Let $(x, y, z)$ be a point on the line. Then, we can express the coordinates of any point on the line as:

$$(x, y, z) = A + t\vec{AB}$$

$$(x, y, z) = (1, 2, 4) + t(-2, 3, -1)$$

This gives us the parametric equations:

$$x = 1 - 2t$$

$$y = 2 + 3t$$

$$z = 4 - t$$

To find the symmetric equation, we solve each equation for $t$:

$$t = \frac{1 - x}{2}$$

$$t = \frac{y - 2}{3}$$

$$t = \frac{4 - z}{1}$$

Setting them equal to each other gives us the symmetric equation:

$$\frac{1 - x}{2} = \frac{y - 2}{3} = \frac{4 - z}{1}$$

Or,

$$\frac{x - 1}{-2} = \frac{y - 2}{3} = \frac{z - 4}{-1}$$

3. Final Answer

The equation of the line AB is $\frac{x - 1}{-2} = \frac{y - 2}{3} = \frac{z - 4}{-1}$.