How to Find x When Three Points Are Collinear in Vector Form

Key concept: When three points lie on one straight line, their position vectors must satisfy a proportional relationship. That gives a clean way to solve for the unknown coordinate.

Step by step

Let

$A=(0,2),\quad B=(4,10),\quad C=(x,14).$

For collinearity, the direction vectors must be parallel.

Step 1: Find $\overrightarrow{AB}$ and $\overrightarrow{BC}$

$\overrightarrow{AB}=(4-0,\,10-2)=(4,8)$

$\overrightarrow{BC}=(x-4,\,14-10)=(x-4,4)$

Step 2: Set the ratios equal

For parallel vectors,

$\frac{4}{x-4}=\frac{8}{4}$

$\frac{4}{x-4}=2$

Step 3: Solve

$4=2(x-4)$

$4=2x-8$

$2x=12$

$x=6$

So the required value is

$\boxed{6}$

Pitfall alert

A common mistake is to compare the position vectors directly and forget that collinearity is about the vectors between points, not just the coordinates themselves. Also watch the sign when forming $\overrightarrow{AB}$ and $\overrightarrow{BC}$.

Try different conditions

If the middle point changed, the same idea still works: form two direction vectors from any two pairs of points, then check whether one is a scalar multiple of the other. If the unknown were in a different coordinate, the final algebra would change, but the collinearity test stays the same.

Further reading

collinear points, position vector, parallel vectors