Question

derivation of distance formula in 3D space

Answer 1

To derive the distance formula in 3D space, we start with the concepts of the Cartesian coordinate system. In 3D space, a point can be represented by its coordinates \((x_1, y_1, z_1)\) and another point by \((x_2, y_2, z_2)\).

The distance \(d\) between these two points can be found using the Euclidean distance formula, which is an extension of the Pythagorean theorem.

### Step 1: Analyze the 3D space

Consider two points:
- Point A: \((x_1, y_1, z_1)\)
- Point B: \((x_2, y_2, z_2)\)

### Step 2: Calculate the differences in each coordinate

First, calculate the differences in the \(x\), \(y\), and \(z\) coordinates:
- \( \Delta x = x_2 - x_1 \)
- \( \Delta y = y_2 - y_1 \)
- \( \Delta z = z_2 - z_1 \)

### Step 3: Use the Pythagorean theorem

We can first think of this distance in terms of a right triangle in two dimensions. In 2D space, if we consider the distance between two points as:
\[
d_{2D} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

In 3D, we can consider the distance \(d\) as a hypotenuse of a right triangle where:
- One leg is the distance in the \(xy\)-plane, and is given by the expression \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
- The other leg is the vertical difference in the \(z\)-direction, which is \(|z_2 - z_1| = |\Delta z|\).

### Step 4: Combine the dimensions

Now, we can combine these two distances to find the overall distance \(d\):
\[
d^2 = \left( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \right)^2 + (z_2 - z_1)^2
\]

This simplifies to:
\[
d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2
\]

### Step 5: Take the square root

To find the distance \(d\), we take the square root of both sides:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

### Final Distance Formula

Thus, the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

This is the derivation of the distance formula in 3D space.