(x₁, y₁, z₁) and (x₂, y₂, z₂) are the coordinates of two points.
m and n are the ratios in which a point divides the line segment.
The formula
d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
x₁, y₁, z₁coordinates of first point
x₂, y₂, z₂coordinates of second point
ddistance between the two points
Let's understand it simply
Imagine you are in a big hall, like a cinema. You want to know how far you are from the screen (distance), where the exact middle seat is (midpoint), or where your friend is sitting if you both agreed to meet at a point that is 1/3rd from the front (section formula). In 3D, you just add the height (z) to your usual left-right (x) and front-back (y) directions.
How it works
Distance Formula: Find the difference in x, y, and z between the two points.
Square each difference.
Add all three squares.
Take the square root of the sum. This gives the straight-line distance.
Midpoint Formula: Add the x, y, and z values of both points and divide each by 2.
Section Formula: Multiply each coordinate by the opposite ratio part, add, and divide by the total ratio.
Finding Distance in 3D
Find x, y, z differences
Square each difference
Add the squares
Take the square root
The story behind it
These formulas come from the Pythagorean theorem, which helps find the shortest distance between two points. Mathematicians extended this idea from flat surfaces (2D) to space (3D) when they needed to solve real-world problems, like navigation and construction.
Where you see it in real life
Cricket stadium: Finding the distance from the pitch to the top of the stadium roof (3D distance).
Elevator in a building: Calculating the straight path from the ground floor corner to a top floor corner (distance formula).
Drone flying: Finding the midpoint between two heights and locations (midpoint formula).
Dividing a ladder: Placing a support at a certain ratio between two ends (section formula).
Mapping cities: Finding the exact spot between two cities at different heights (midpoint/section formula).
Why it matters
Used in engineering to design buildings and bridges.
Needed in computer graphics for 3D games and movies.
Important for navigation (airplanes, ships, drones).
Questions on these formulas come in board exams and entrance tests.
Worked example
Given: A(1, 2, 3), B(4, 6, 8)
Step 1: Write the distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Step 2: Put in the numbers: d = √((4-1)² + (6-2)² + (8-3)²)
Step 3: Calculate: d = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50
Step 4: Answer: d ≈ 7.07 units
Common mistakes
Forgetting to include the z-coordinate in 3D problems.
Mixing up which point is x₁, y₁, z₁ and which is x₂, y₂, z₂ (it doesn’t matter for distance, but matters for section formula if the ratio is not 1:1).
Not squaring the differences before adding in the distance formula.
Using the 2D formula by mistake for 3D questions.
Remember it
Think: "3D means three differences squared and square-rooted for distance!" For midpoint, just average all three coordinates. For section, multiply and add each coordinate with the ratio, then divide by total.
Check yourself
What is the main difference between the 2D and 3D distance formula?
The 3D formula includes the z-coordinate difference squared.
Why do we square the differences in the distance formula?
To make all differences positive and use the Pythagorean theorem.
If you forget the z-coordinate in a 3D problem, what mistake happens?
You get the wrong (shorter) distance, as you ignore height/depth.
Quick recap
3D formulas add the z-coordinate to 2D formulas.
Distance formula: square root of sum of squares of differences.
Midpoint: average all three coordinates.
Section formula: use the ratio to split each coordinate.
Always check if the problem is 3D before choosing the formula.
