Collinear Points Using Vectors

Vectors prove collinearity through scalar multiples: if $\overrightarrow{AB}$ is a multiple of $\overrightarrow{AC}$, the segments are parallel, and sharing the point $A$ makes the three points collinear. It mirrors the gradient method but in vector form.

Points $A$, $B$ and $C$ have position vectors $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$, $\mathbf{b} = 4\mathbf{i} + 7\mathbf{j}$ and $\mathbf{c} = 7\mathbf{i} + 13\mathbf{j}$. Show that $A$, $B$ and $C$ are collinear. [4]

The working

Step 1, find $\overrightarrow{AB}$ (destination minus origin, $\mathbf{b} - \mathbf{a}$): $\overrightarrow{AB} = (4\mathbf{i} + 7\mathbf{j}) - (2\mathbf{i} + 3\mathbf{j}) = 2\mathbf{i} + 4\mathbf{j} \quad \text{(M1)}$

Step 2, find $\overrightarrow{AC}$ ($\mathbf{c} - \mathbf{a}$): $\overrightarrow{AC} = (7\mathbf{i} + 13\mathbf{j}) - (2\mathbf{i} + 3\mathbf{j}) = 5\mathbf{i} + 10\mathbf{j} \quad \text{(M1)}$

Step 3, show one is a scalar multiple of the other: $\overrightarrow{AC} = 5\mathbf{i} + 10\mathbf{j} = \frac{5}{2}(2\mathbf{i} + 4\mathbf{j}) = \frac{5}{2}\,\overrightarrow{AB} \quad \text{(A1)}$

Step 4, conclude. Since $\overrightarrow{AC} = \frac{5}{2}\overrightarrow{AB}$, the vectors are parallel, and they share the point $A$, so $A$, $B$, $C$ are collinear. $\blacksquare$ (A1)

Where the marks are won and lost

• Find two displacement vectors from a common point ($\overrightarrow{AB}$ and $\overrightarrow{AC}$, both from $A$).
• Show one is a scalar multiple of the other, that’s the parallel condition. Here $\frac{5}{2}$ works for both components; check both.
• The shared point is essential to the conclusion, parallel plus a common point equals collinear. State it.

Common mistakes

• Computing $\mathbf{a} - \mathbf{b}$ (wrong direction) for the displacement.
• Finding the scalar for the $\mathbf{i}$-components but not checking the $\mathbf{j}$-components match the same multiple.
• Concluding “parallel” without noting the common point (so not fully “collinear”).

Full method: Vector Problem-Solving notes. See also Position Vectors. Topic home: Vectors pillar.