❖ In this lesson, we’ll learn the distance formula in \(3D\) space. We’ll calculate the distance and then build the formula visually step by step.

The distance formula is one of the most important topics in coordinate geometry, and it has many real‑world applications. In \(3D\), points lie on a coordinate system (\(xyz\)-space).

📌 Distance Formula (\(3D\))

If \(P_1=(x_1, y_1, z_1)\) and \(P_2=(x_2, y_2, z_2)\), then:

$$d = \sqrt{(x_2 − x_1)^2 + (y_2 − y_1)^2 + (z_2 − z_1)^2}$$

So, the distance between two points in \(3D\) is the square root of the sum of the squares of the differences in \(x, y,\) and \(z\).

💡 Key Idea

The \(3D\) distance formula comes from applying the Pythagorean Theorem twice, first in the \(xy\)-plane, then using the \(z\)-direction.

🧭 Direction shortcut:

🔵 \(x\): + →, − ←

🔴 \(y\): + forward, − backward

🟢 \(z\): + ↑, − ↓

✅ Perfect for students learning coordinate geometry with clear steps and easy visual explanations.