Q.Show that the three points $A(2,3,4)$, $B(-1,2,-3)$ and $C(-4,1,-10)$ are collinear and find the ratio in which C divides AB.

Collinearity Slope Condition — From Intuition to Precision

Imagine three points scattered on a sheet of paper. You can always draw a triangle through them. But if those three points happen to lie perfectly on a single straight line — like beads on a thread — they are called collinear (from Latin co- meaning "together" and linearis meaning "belonging to a line").

The question is: how do you check, using only coordinates, whether three given points are collinear?

The Intuition

Think about walking from point A to point B, then from point B to point C. If all three lie on the same line, your direction of travel should not change when you turn at B. In other words, the slope of AB must equal the slope of BC.

Slope measures steepness: $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$. If two segments share the same slope and meet at a common point (B), they lie on the same straight line.

The Precise Statement

Let three points be $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$. They are collinear if and only if:

provided that $x_1 \neq x_2$ and $x_2 \neq x_3$ (i.e., no vertical segment).

What About Vertical Lines?

If $x_1 = x_2$, the slope of AB is undefined (division by zero). But the condition still works: if AB is vertical, then for collinearity, BC must also be vertical — meaning $x_2 = x_3$. So the condition becomes: either both slopes are equal and defined, or both are undefined (i.e., both segments are vertical).

A Cleaner Algebraic Form

Cross-multiplying the slope equality gives a form that avoids division entirely:

This works for all cases, including vertical lines.

Example

Check if $A(1, 2)$, $B(3, 6)$, $C(5, 10)$ are collinear.

Slope of AB: $\frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$

Slope of BC: $\frac{10 - 6}{5 - 3} = \frac{4}{2} = 2$

Slopes are equal → points are collinear. Indeed, they all lie on $y = 2x$.

Why This Matters

The slope condition is the simplest test for collinearity in coordinate geometry. It appears in: …