Q.Show that the points $A(1, -2, -8)$, $B(5, 0, -2)$ and $C(11, 3, 7)$ are collinear, and find the ratio in which B divides AC.

Show $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel, then read the ratio.

Step 1 — Vectors from $A$.

Step 2 — Check parallelism. $\dfrac{4}{10}=\dfrac{2}{5},\ \dfrac{2}{5}=\dfrac{2}{5},\ \dfrac{6}{15}=\dfrac{2}{5}.$ All equal, so $\overrightarrow{AB} = \dfrac{2}{5}\overrightarrow{AC}$; the three points are collinear. …

The idea

Three points lie on one straight line when the vector from a common point to the second is a scalar multiple of the vector to the third: if $\overrightarrow{AB} = \lambda\,\overrightarrow{AC}$, the points are collinear. The value of $\lambda$ then tells us where $B$ sits along $AC$ and hence the division ratio.

Step-by-step

1. Form two vectors from $A$.

2. Test for a common ratio.

All three match, so $\overrightarrow{AB} = \dfrac{2}{5}\overrightarrow{AC}$. A scalar multiple exists, hence $A$, $B$, $C$ are collinear. …

Method: Proving Collinearity and Finding the Division Ratio

Use this to show three points lie on a line and to locate how the middle point divides the segment.

Steps

Step 1: Build two vectors from a common point

From $A$, form $\overrightarrow{AB}=B-A$ and $\overrightarrow{AC}=C-A$ (component-wise, head minus tail).

Step 2: Test for a scalar multiple

The points are collinear iff $\overrightarrow{AB}=\lambda\,\overrightarrow{AC}$, i.e. the components share a common ratio:

$\frac{(AB)_x}{(AC)_x}=\frac{(AB)_y}{(AC)_y}=\frac{(AB)_z}{(AC)_z}=\lambda.$ …

Common Mistakes

Mistake 1: Testing equal lengths instead of equal direction

Why it's wrong: collinearity is about direction — $\overrightarrow{AB}$ being a scalar multiple of $\overrightarrow{AC}$ — not about $|\overrightarrow{AB}|=|\overrightarrow{AC}|$. Correct approach: check the component ratios are all equal.

Mistake 2: Misreading $\lambda$ as the ratio directly …