Find the length of the perpendicular drawn from the point P (3, -4, 5) on the z-axis.

Answer

To find the length of the perpendicular drawn from the point P(3, -4, 5) on the z-axis, we can follow these steps:

### Step 1: Understand the Position of the Point
The point P(3, -4, 5) has coordinates where:
- x = 3
- y = -4
- z = 5

The z-axis is defined by the points where x = 0 and y = 0, meaning any point on the z-axis can be represented as (0, 0, z).

### Step 2: Identify the Projection on the Z-axis
To find the perpendicular distance from point P to the z-axis, we need to consider the projection of point P onto the z-axis. The projection will have the same z-coordinate as point P but will have x and y coordinates equal to zero. Therefore, the projection of P onto the z-axis is (0, 0, 5).

### Step 3: Calculate the Distance
The distance between point P(3, -4, 5) and its projection on the z-axis (0, 0, 5) can be calculated using the distance formula in three dimensions. The distance \( D \) is given by:

\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the coordinates:
- \( (x_1, y_1, z_1) = (3, -4, 5) \)
- \( (x_2, y_2, z_2) = (0, 0, 5) \)

The distance becomes:

\[ D = \sqrt{(0 - 3)^2 + (0 - (-4))^2 + (5 - 5)^2} \]

### Step 4: Simplify the Expression
Calculating the squares:

\[ D = \sqrt{(-3)^2 + (4)^2 + (0)^2} \]
\[ D = \sqrt{9 + 16 + 0} \]
\[ D = \sqrt{25} \]

### Step 5: Final Calculation
Taking the square root gives:

\[ D = 5 \]

Thus, the length of the perpendicular drawn from the point P(3, -4, 5) to the z-axis is **5 units**.

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