Finding λ So That Two Vectors Are Parallel

Two vectors are parallel when one is a scalar multiple of the other, equivalently, their components are in the same ratio. Setting the ratios equal gives the unknown.

The vectors $\mathbf{a} = 3\mathbf{i} + 2\mathbf{j}$ and $\mathbf{b} = \lambda\mathbf{i} + 4\mathbf{j}$ are parallel. Find the value of $\lambda$. [3]

The working

Step 1, write the parallel condition. $\mathbf{b} = k\mathbf{a}$ for some scalar $k$, so the components are in the same ratio: $\frac{\lambda}{3} = \frac{4}{2} \quad \text{(M1)}$

Step 2, evaluate the known ratio ($\frac{4}{2} = 2$, so $\mathbf{b} = 2\mathbf{a}$): $\frac{\lambda}{3} = 2 \quad \text{(M1)}$

Step 3, solve for $\lambda$: $\lambda = 6 \quad \text{(A1)}$

Check: $\mathbf{b} = 6\mathbf{i} + 4\mathbf{j} = 2(3\mathbf{i} + 2\mathbf{j}) = 2\mathbf{a}$ ✓, confirming they’re parallel.

Where the marks are won and lost

• Parallel means scalar multiple, so the components share a common ratio: $\frac{\lambda}{3} = \frac{4}{2}$. Match the $\mathbf{i}$-ratio to the $\mathbf{j}$-ratio.
• The scalar $k = 2$ comes from the known components ($\frac42$); apply the same $k$ to the unknown component.
• A quick check ($\mathbf{b} = 2\mathbf{a}$) confirms the answer.

Common mistakes

• Setting the components equal ($\lambda = 4$) instead of proportional.
• Inverting the ratio ($\frac{3}{\lambda} = \frac{4}{2}$) and getting $\lambda = \frac32$.
• Forgetting that parallel allows any scalar, not just $1$.

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