Findthe distance of `P(a,b,c)` from x,y and z-axes.

Explanation

To find the distance of the point \( P(a, b, c) \) from the x-axis, y-axis, and z-axis, we can follow these steps:

### Step 1: Distance from the x-axis
The distance from a point \( P(a, b, c) \) to the x-axis can be found by considering the projection of point \( P \) onto the x-axis. The coordinates of the point on the x-axis that is closest to \( P \) will be \( (a, 0, 0) \).

The distance \( d_x \) from point \( P \) to the x-axis is given by the formula:
\[ d_x = \sqrt{(a - a)^2 + (b - 0)^2 + (c - 0)^2} = \sqrt{0 + b^2 + c^2} = \sqrt{b^2 + c^2} \]

### Step 2: Distance from the y-axis
Next, we find the distance from point \( P(a, b, c) \) to the y-axis. The closest point on the y-axis to \( P \) will have coordinates \( (0, b, 0) \).

The distance \( d_y \) from point \( P \) to the y-axis is given by:
\[ d_y = \sqrt{(a - 0)^2 + (b - b)^2 + (c - 0)^2} = \sqrt{a^2 + 0 + c^2} = \sqrt{a^2 + c^2} \]

### Step 3: Distance from the z-axis
Finally, we calculate the distance from point \( P(a, b, c) \) to the z-axis. The closest point on the z-axis to \( P \) will have coordinates \( (0, 0, c) \).

The distance \( d_z \) from point \( P \) to the z-axis is given by:
\[ d_z = \sqrt{(a - 0)^2 + (b - 0)^2 + (c - c)^2} = \sqrt{a^2 + b^2 + 0} = \sqrt{a^2 + b^2} \]

### Final Results
Thus, the distances from the point \( P(a, b, c) \) to the three axes are:
- Distance from the x-axis: \( \sqrt{b^2 + c^2} \)
- Distance from the y-axis: \( \sqrt{a^2 + c^2} \)
- Distance from the z-axis: \( \sqrt{a^2 + b^2} \)