Expressing a Vector as a Combination of Two Others

To write one vector as a combination of two others, compare components. The $\mathbf{i}$-parts must match and the $\mathbf{j}$-parts must match, giving two simultaneous equations in the scalars $\lambda$ and $\mu$.

The vectors $\mathbf{a} = \mathbf{i} + \mathbf{j}$, $\mathbf{b} = \mathbf{i} - \mathbf{j}$ and $\mathbf{c} = 5\mathbf{i} + \mathbf{j}$ are given. Find scalars $\lambda$ and $\mu$ such that $\mathbf{c} = \lambda\mathbf{a} + \mu\mathbf{b}$. [5]

The working

Step 1, write out $\lambda\mathbf{a} + \mu\mathbf{b}$ and group the components: $\lambda(\mathbf{i} + \mathbf{j}) + \mu(\mathbf{i} - \mathbf{j}) = (\lambda + \mu)\mathbf{i} + (\lambda - \mu)\mathbf{j} \quad \text{(M1)}$

Step 2, compare with $\mathbf{c} = 5\mathbf{i} + \mathbf{j}$, matching $\mathbf{i}$ and $\mathbf{j}$ components: $\lambda + \mu = 5 \qquad (\mathbf{i}\text{-components})$ $\lambda - \mu = 1 \qquad (\mathbf{j}\text{-components}) \quad \text{(M1, A1)}$

Step 3, solve simultaneously. Add the two equations (the $\mu$ cancels): $2\lambda = 6 \;\Rightarrow\; \lambda = 3 \quad \text{(M1)}$

Substitute back into $\lambda + \mu = 5$: $3 + \mu = 5 \;\Rightarrow\; \mu = 2 \quad \text{(A1)}$

So $\mathbf{c} = 3\mathbf{a} + 2\mathbf{b}$. Check: $3(\mathbf{i}+\mathbf{j}) + 2(\mathbf{i}-\mathbf{j}) = 5\mathbf{i} + \mathbf{j} = \mathbf{c}$ ✓.

Where the marks are won and lost

• Comparing components is the method: two vectors are equal only if both components agree, giving one equation each.
• Keep the $\mathbf{i}$-equation and $\mathbf{j}$-equation straight: $\lambda + \mu = 5$ (from $\mathbf{i}$) and $\lambda - \mu = 1$ (from $\mathbf{j}$). Swapping them scrambles the solution.
• The check by substitution is quick insurance and is often rewarded.

Common mistakes

• Failing to expand $\mu(\mathbf{i} - \mathbf{j})$ to $\mu\mathbf{i} - \mu\mathbf{j}$ (sign on the $\mathbf{j}$).
• Setting up only one equation and guessing the other scalar.
• Arithmetic slips solving the simultaneous pair.

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