Find the ratio in which yz-plane divides the line segment joining the points P(-1, 3,2) and Q(3, -4,5). Also find the co-ordinates of point of division.

Explanation

To find the ratio in which the yz-plane divides the line segment joining the points P(-1, 3, 2) and Q(3, -4, 5), we can follow these steps:

### Step 1: Understand the yz-plane
The yz-plane is defined by the equation x = 0. This means that any point on the yz-plane will have its x-coordinate equal to 0.

### Step 2: Use the section formula
Let the point of division R divide the line segment PQ in the ratio m:n. The coordinates of R can be expressed using the section formula as follows:
- \( R_x = \frac{n \cdot x_1 + m \cdot x_2}{m + n} \)
- \( R_y = \frac{n \cdot y_1 + m \cdot y_2}{m + n} \)
- \( R_z = \frac{n \cdot z_1 + m \cdot z_2}{m + n} \)

Where:
- \( P(x_1, y_1, z_1) = (-1, 3, 2) \)
- \( Q(x_2, y_2, z_2) = (3, -4, 5) \)

### Step 3: Set the x-coordinate of R to 0
Since R lies on the yz-plane, we set \( R_x = 0 \):
\[ 0 = \frac{n \cdot (-1) + m \cdot 3}{m + n} \]
This simplifies to:
\[ 0 = -n + 3m \implies 3m = n \implies \frac{m}{n} = \frac{1}{3} \]
Thus, the ratio \( m:n = 1:3 \).

### Step 4: Find the coordinates of point R
Now we can find the y and z coordinates of R using the ratio \( m:n = 1:3 \).

#### Finding R_y:
\[ R_y = \frac{n \cdot y_1 + m \cdot y_2}{m + n} = \frac{3 \cdot 3 + 1 \cdot (-4)}{1 + 3} = \frac{9 - 4}{4} = \frac{5}{4} \]

#### Finding R_z:
\[ R_z = \frac{n \cdot z_1 + m \cdot z_2}{m + n} = \frac{3 \cdot 2 + 1 \cdot 5}{1 + 3} = \frac{6 + 5}{4} = \frac{11}{4} \]

### Step 5: Conclusion
Thus, the coordinates of point R, where the yz-plane divides the line segment joining P and Q, are:
\[ R(0, \frac{5}{4}, \frac{11}{4}) \]
And the ratio in which the yz-plane divides the segment is \( 1:3 \).