How to add and subtract vectors

This method combines two or more vectors into a single vector by adding or subtracting them. In the Cartesian form $a\mathbf{i} + b\mathbf{j}$, or the column form $\begin{pmatrix} a \\ b \end{pmatrix}$, you simply add or subtract the matching components.

Geometrically, addition follows the triangle law (place the vectors head to tail and join the ends), and subtraction is just adding the negative, $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$. It answers questions such as 'find $\mathbf{a} + \mathbf{b}$', 'find the resultant', or 'given the position vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$, express $\overrightarrow{AB}$ in terms of $\mathbf{i}$ and $\mathbf{j}$'.

It is the foundation the rest of the Vectors chapter is built on, resultants, collinearity, ratios and magnitudes all start here.

Reach for this method whenever vectors are given in component form or as position vectors and the question asks for a sum, a difference, a resultant, or a directed line segment. Signals include 'find $\mathbf{a} + \mathbf{b}$', 'the resultant of the two vectors', 'express $\overrightarrow{AB}$ in terms of $\mathbf{i}$ and $\mathbf{j}$', or a diagram of arrows placed head to tail.

When you are given position vectors from an origin $O$, any directed segment between two points is a subtraction: $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$. Reading this correctly, end point minus start point, is the move that decides whether your answer points the right way.

Once the target is written as a sum or difference of known vectors, the rest is componentwise arithmetic.

1. 1

   Write each vector in component form

   Express every vector as $a\mathbf{i} + b\mathbf{j}$ (or a column vector) so the $\mathbf{i}$ and $\mathbf{j}$ parts line up.

2. 2

   Decide add or subtract

   For a resultant, add. For a directed segment $\overrightarrow{AB}$, use $\overrightarrow{OB} - \overrightarrow{OA}$, end minus start.

3. 3

   Combine the i-components

   Add or subtract the numbers in front of $\mathbf{i}$, keeping track of signs.

4. 4

   Combine the j-components

   Do the same for the numbers in front of $\mathbf{j}$; subtracting a negative adds.

5. 5

   Write the single result

   State the answer as one vector $p\mathbf{i} + q\mathbf{j}$ (or a column vector).

6. 6

   Check with the triangle law

   For a segment, confirm $\overrightarrow{OA} + \overrightarrow{AB} = \overrightarrow{OB}$.

• Combining an $\mathbf{i}$-component with a $\mathbf{j}$-component, the two directions must be kept completely separate.
• Sign errors when subtracting a negative: $2 - (-4) = 6$, not $-2$.
• Writing $\overrightarrow{AB} = \overrightarrow{OA} - \overrightarrow{OB}$, which points the wrong way; it is end minus start, $\overrightarrow{OB} - \overrightarrow{OA}$.
• Dropping a term when one component is zero, instead of writing it explicitly.
• Mixing column-vector and $\mathbf{i}$–$\mathbf{j}$ notation midway, which scrambles the bookkeeping.

The single step that trips students is the direction of a subtraction, writing $\overrightarrow{OA} - \overrightarrow{OB}$ when $\overrightarrow{AB}$ needs $\overrightarrow{OB} - \overrightarrow{OA}$, so the answer comes out with every sign reversed. In one-to-one lessons our teachers give you the fixed phrase 'end minus start' and make you write it beside every directed segment before you touch the numbers.

We stack the $\mathbf{i}$ parts and $\mathbf{j}$ parts in separate columns so a $2 - (-4)$ can never quietly become $-2$, and we finish with the triangle-law check you saw above. Because SPM Add Math uses analytic marking, setting up $\overrightarrow{OB} - \overrightarrow{OA}$ correctly already earns method marks even if a sign slips later.

Note that lessons are taught in English, though the SPM paper itself is set bilingually. Every teacher on spmaddmath.com.my is experienced.

Lessons are online, at your own pace.