Planes are drawn parallel to the coordinate planes through the points `(1, 2,3) and (3, -4, -5).` Find th lengths of the edges of the parallelopiped so formed.

Answer

To find the lengths of the edges of the parallelepiped formed by the planes drawn parallel to the coordinate planes through the points \( (1, 2, 3) \) and \( (3, -4, -5) \), we will follow these steps:

### Step 1: Identify the Points
Let the points be:
- Point A: \( (1, 2, 3) \)
- Point B: \( (3, -4, -5) \)

### Step 2: Determine the Planes
Planes parallel to the coordinate planes through point A are:
1. \( x = 1 \)
2. \( y = 2 \)
3. \( z = 3 \)

Planes parallel to the coordinate planes through point B are:
1. \( x = 3 \)
2. \( y = -4 \)
3. \( z = -5 \)

### Step 3: Calculate Lengths Along Each Axis
To find the lengths of the edges of the parallelepiped, we will calculate the distances along the x-axis, y-axis, and z-axis.

#### Length along the x-axis:
The length along the x-axis is given by the absolute difference between the x-coordinates of points A and B:
\[ \text{Length along x-axis} = |3 - 1| = 2 \text{ units} \]

#### Length along the y-axis:
The length along the y-axis is given by the absolute difference between the y-coordinates of points A and B:
\[ \text{Length along y-axis} = |-4 - 2| = |-6| = 6 \text{ units} \]

#### Length along the z-axis:
The length along the z-axis is given by the absolute difference between the z-coordinates of points A and B:
\[ \text{Length along z-axis} = |-5 - 3| = |-8| = 8 \text{ units} \]

### Step 4: Summarize the Lengths
Thus, the lengths of the edges of the parallelepiped formed are:
- Length along the x-axis: \( 2 \) units
- Length along the y-axis: \( 6 \) units
- Length along the z-axis: \( 8 \) units

### Final Answer
The lengths of the edges of the parallelepiped are:
- \( 2 \) units (x-axis)
- \( 6 \) units (y-axis)
- \( 8 \) units (z-axis)

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