Find the perpendicualr distances of the point P(a,b,c) form the co - ordinate axes

Answer

To find the perpendicular distances of the point \( P(a, b, c) \) from the coordinate axes, we will calculate the distances from the x-axis, y-axis, and z-axis separately.

### Step-by-Step Solution:

1. **Distance from the x-axis:**
- The coordinates of the point \( P \) are \( (a, b, c) \).
- The corresponding point on the x-axis is \( (a, 0, 0) \).
- The formula for the distance between two points in 3D is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
- For the distance from point \( P(a, b, c) \) to the x-axis point \( (a, 0, 0) \):
\[ d_x = \sqrt{(a - a)^2 + (b - 0)^2 + (c - 0)^2} = \sqrt{0 + b^2 + c^2} = \sqrt{b^2 + c^2} \]

2. **Distance from the y-axis:**
- The corresponding point on the y-axis is \( (0, b, 0) \).
- For the distance from point \( P(a, b, c) \) to the y-axis point \( (0, b, 0) \):
\[ d_y = \sqrt{(0 - a)^2 + (b - b)^2 + (0 - c)^2} = \sqrt{a^2 + 0 + c^2} = \sqrt{a^2 + c^2} \]

3. **Distance from the z-axis:**
- The corresponding point on the z-axis is \( (0, 0, c) \).
- For the distance from point \( P(a, b, c) \) to the z-axis point \( (0, 0, c) \):
\[ d_z = \sqrt{(0 - a)^2 + (0 - b)^2 + (c - c)^2} = \sqrt{a^2 + b^2 + 0} = \sqrt{a^2 + b^2} \]

### Final Results:
- The perpendicular distance from the x-axis is \( \sqrt{b^2 + c^2} \).
- The perpendicular distance from the y-axis is \( \sqrt{a^2 + c^2} \).
- The perpendicular distance from the z-axis is \( \sqrt{a^2 + b^2} \).