To find the distance between the point \( P(2m, 3m, 4m) \) and the \( x \)-axis, we can follow these steps:
### Step 1: Understand the coordinates of the point and the x-axis
The point \( P \) has coordinates \( (2m, 3m, 4m) \). The \( x \)-axis is defined by the points where \( y = 0 \) and \( z = 0 \). Therefore, any point on the \( x \)-axis can be represented as \( (x, 0, 0) \).
### Step 2: Identify the coordinates of the point on the x-axis
To find the distance from point \( P \) to the \( x \)-axis, we can consider the point on the \( x \)-axis that is directly below \( P \). This point will have the same \( x \)-coordinate as \( P \) but \( y \) and \( z \) coordinates equal to 0. Thus, the coordinates of the point on the \( x \)-axis are \( (2m, 0, 0) \).
### Step 3: Use the distance formula
The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in three-dimensional space is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
In our case, we will use the coordinates of point \( P(2m, 3m, 4m) \) and the point on the \( x \)-axis \( (2m, 0, 0) \).
### Step 4: Substitute the coordinates into the distance formula
Let:
- \( (x_1, y_1, z_1) = (2m, 3m, 4m) \)
- \( (x_2, y_2, z_2) = (2m, 0, 0) \)
Substituting these values into the distance formula:
\[
d = \sqrt{(2m - 2m)^2 + (0 - 3m)^2 + (0 - 4m)^2}
\]
### Step 5: Simplify the expression
Calculating each term:
- The first term: \( (2m - 2m)^2 = 0^2 = 0 \)
- The second term: \( (0 - 3m)^2 = (-3m)^2 = 9m^2 \)
- The third term: \( (0 - 4m)^2 = (-4m)^2 = 16m^2 \)
Now, substituting back into the equation:
\[
d = \sqrt{0 + 9m^2 + 16m^2} = \sqrt{25m^2}
\]
### Step 6: Calculate the final distance
Taking the square root:
\[
d = 5m
\]
### Conclusion
The distance between the point \( P(2m, 3m, 4m) \) and the \( x \)-axis is \( 5m \).
