Position Vectors and the Displacement Between Points

The core rule of position vectors is “destination minus origin”: $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$. Get the subtraction order right and the displacement, and then its length, follow directly.

Points $A$ and $B$ have position vectors $\mathbf{a} = 2\mathbf{i} + \mathbf{j}$ and $\mathbf{b} = 5\mathbf{i} + 7\mathbf{j}$ relative to an origin $O$. (i) Find $\overrightarrow{AB}$. [2] (ii) Find $\left|\overrightarrow{AB}\right|$, giving your answer in exact (surd) form. [2]

The working

(i) Displacement is destination minus origin, $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$: $\overrightarrow{AB} = (5\mathbf{i} + 7\mathbf{j}) - (2\mathbf{i} + \mathbf{j}) = 3\mathbf{i} + 6\mathbf{j} \quad \text{(M1, A1)}$

(ii) Its magnitude: $\left|\overrightarrow{AB}\right| = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \quad \text{(M1)}$

Simplify the surd ($45 = 9 \times 5$): $\sqrt{45} = \sqrt{9}\,\sqrt{5} = 3\sqrt{5} \quad \text{(A1)}$

Where the marks are won and lost

• $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$, not $\mathbf{a} - \mathbf{b}$. The wrong order gives $\overrightarrow{BA}$ (the reverse vector).
• Subtract component by component: $(5 - 2)\mathbf{i} + (7 - 1)\mathbf{j}$. The $\mathbf{j}$-component of $\mathbf{a}$ is $1$ (from the bare $\mathbf{j}$), easy to overlook.
• Exact form means simplifying $\sqrt{45}$ to $3\sqrt{5}$. Leaving $\sqrt{45}$ or giving $6.7$ can lose the accuracy mark.

Common mistakes

• Computing $\mathbf{a} - \mathbf{b}$ and getting the direction reversed.
• Treating the coefficient of $\mathbf{j}$ in $\mathbf{a}$ as $0$ instead of $1$.
• Not simplifying the surd when exact form is required.

Full method: Position Vectors notes. Topic home: Vectors pillar.