Find the ratio in which the line segment having the end points `A(-1, -3, 4)` and `B(4, 2, -1) ` is divided by the `xz-`plane. Also, find the coordinates of the point of division.

Explanation

To solve the problem of finding the ratio in which the line segment with endpoints \( A(-1, -3, 4) \) and \( B(4, 2, -1) \) is divided by the \( xz \)-plane, and to find the coordinates of the point of division, we can follow these steps:

### Step 1: Understand the \( xz \)-plane
The \( xz \)-plane is defined by the equation \( y = 0 \). This means that any point on the \( xz \)-plane will have a \( y \)-coordinate of 0.

### Step 2: Set up the section formula
Let the point of intersection of the line segment \( AB \) with the \( xz \)-plane be \( P(x, 0, z) \). We will use the section formula to find the coordinates of point \( P \) in terms of a ratio \( k:1 \), where \( k \) is the ratio in which the segment is divided.

### Step 3: Apply the section formula for the \( y \)-coordinate
Using the section formula for the \( y \)-coordinate, we have:
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]
Here, \( m = k \), \( n = 1 \), \( y_1 = -3 \) (from point \( A \)), and \( y_2 = 2 \) (from point \( B \)). Setting \( y = 0 \) (since it lies on the \( xz \)-plane), we get:
\[ 0 = \frac{k \cdot 2 + 1 \cdot (-3)}{k + 1} \]
This simplifies to:
\[ 0 = \frac{2k - 3}{k + 1} \]
From this, we can conclude that:
\[ 2k - 3 = 0 \implies 2k = 3 \implies k = \frac{3}{2} \]

### Step 4: Determine the ratio
The ratio in which the line segment is divided is \( k:1 = \frac{3}{2}:1 \), which simplifies to \( 3:2 \).

### Step 5: Find the coordinates of point \( P \)
Now, we will find the \( x \) and \( z \) coordinates of point \( P \) using the section formula.

#### For the \( x \)-coordinate:
Using the section formula for the \( x \)-coordinate:
\[ x = \frac{k \cdot x_2 + n \cdot x_1}{k+n} \]
where \( x_1 = -1 \) and \( x_2 = 4 \):
\[ x = \frac{\frac{3}{2} \cdot 4 + 1 \cdot (-1)}{\frac{3}{2} + 1} = \frac{6 - 1}{\frac{5}{2}} = \frac{5}{\frac{5}{2}} = 2 \]

#### For the \( z \)-coordinate:
Using the section formula for the \( z \)-coordinate:
\[ z = \frac{k \cdot z_2 + n \cdot z_1}{k+n} \]
where \( z_1 = 4 \) and \( z_2 = -1 \):
\[ z = \frac{\frac{3}{2} \cdot (-1) + 1 \cdot 4}{\frac{3}{2} + 1} = \frac{-\frac{3}{2} + 4}{\frac{5}{2}} = \frac{\frac{5}{2}}{\frac{5}{2}} = 1 \]

### Final Result
The coordinates of point \( P \) are \( (2, 0, 1) \).

### Summary
1. The ratio in which the line segment is divided by the \( xz \)-plane is \( 3:2 \).
2. The coordinates of the point of division are \( (2, 0, 1) \).