Vector Problem-Solving

The last marks in a 0606 vectors question are strategy marks: the tools are $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$, magnitudes and ratios, but the question is a multi-step geometric argument. Here is the strategy layer.

Habit 1. Draw and label, always

Every vector geometry question gets a diagram: points labelled, given vectors marked ($\mathbf{a}$, $\mathbf{b}$), ratios annotated on the segments. Not optional decoration, the diagram is where you see that $\overrightarrow{OX}$ can be written two ways, which is the entire solution. Thirty seconds, every time, even when the question provides a figure (re-mark it with what you know).

Habit 2. Write routes

Any vector equals any path between its endpoints: $\overrightarrow{PQ} = \overrightarrow{PO} + \overrightarrow{OQ} = -\mathbf{p} + \mathbf{q}$. When a target vector isn’t directly reachable, write it as a route through known points, and write the route symbolically first, numbers second:

$\overrightarrow{XY} = \overrightarrow{XA} + \overrightarrow{AB} + \overrightarrow{BY}$ ← the route (M mark territory) $= -\frac{1}{3}\mathbf{a} + (\mathbf{b} - \mathbf{a}) + \frac{1}{2}(\mathbf{c} - \mathbf{b})$ ← substitution

Choosing a sensible route through the origin or through ratio points is the puzzle; the route line is how the examiner follows (and credits) your choice.

Habit 3. The two-expressions-equate pattern

The standard hard question: $X$ lies on $OC$ and on $AB$, so $\overrightarrow{OX}$ has two expressions, one with unknown $\lambda$, one with $\mu$. Equate them, then equate coefficients of $\mathbf{a}$ and $\mathbf{b}$ separately, stating “since $\mathbf{a}$ and $\mathbf{b}$ are non-parallel”. Two equations, two unknowns, solve, substitute back, convert $\lambda/\mu$ into the ratio the question asked for. The justification sentence is a mark; the conversion to a stated ratio ("$OX : XC = 3 : 2$") is another, finish the sentence, not just the algebra.

Habit 4. Present proofs as arguments

“Show that $OACB$ is a parallelogram” wants: the relevant vectors computed, the defining property named ($\overrightarrow{OA} = \overrightarrow{BC}$, i.e. one pair of opposite sides equal and parallel), and the conclusion in words. Compute → name the property → conclude. Mark schemes allocate a mark to each stage; fused or missing stages drop them even over correct arithmetic.

Common mistakes

• No diagram, so the two-expression structure is never spotted
• Routes assembled with sign errors (each backwards leg negates)
• Coefficients equated without the non-parallel statement
• $\lambda$ and $\mu$ found but the asked-for ratio never stated
• Proof conclusions left implicit

Full topic context: Vectors notes, and this is the second of the two classically avoided topics (P&C is the other), making it reliable A* differentiation.