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11.2PureCore

Representing Vectors

Vectors are written numerically in two equivalent ways: column vectors and i–j (component) form. Working in components turns vector addition into simple arithmetic on the numbers.

25 min Vectors
Edexcel AS Level Maths: 11.2 Representing Vectors (Column / i, j)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Write vectors in column and i–j form
  • Convert between the two forms
  • Add and subtract vectors in component form
  • Multiply a component vector by a scalar
1

Two notations

A column vector stacks the horizontal and vertical components. The same vector in i\mathbf{i}–j\mathbf{j} form uses unit vectors i\mathbf{i} (one unit right) and j\mathbf{j} (one unit up).

(32)=3i+2j\begin{pmatrix} 3 \\ 2 \end{pmatrix} = 3\mathbf{i} + 2\mathbf{j}
Top number = i\mathbf{i} component; bottom = j\mathbf{j} component.
2

Adding in components

In component form, add (or subtract) the matching components separately. No diagrams needed.

(ab)+(cd)=(a+cb+d)\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a + c \\ b + d \end{pmatrix}
Combine like components.
Example 1Find (2i+3j)+(4i−j)(2\mathbf{i} + 3\mathbf{j}) + (4\mathbf{i} - \mathbf{j}).
1Add i\mathbf{i} parts: 2+4=62 + 4 = 6.
2Add j\mathbf{j} parts: 3−1=23 - 1 = 2.
Answer6i+2j6\mathbf{i} + 2\mathbf{j}
3

Scalar multiplication

To multiply by a scalar, multiply each\textbf{each} component.

Example 1Find 3(2−1)3\begin{pmatrix} 2 \\ -1 \end{pmatrix}.
1Multiply each component by 3.
Answer(6−3)\begin{pmatrix} 6 \\ -3 \end{pmatrix}

Tip — Keep i with i and j with j — never mix the two components.

Formula recap

(ab)=ai+bj\begin{pmatrix} a \\ b \end{pmatrix} = a\mathbf{i} + b\mathbf{j}
Column ↔ i–j form.
(ab)+(cd)=(a+cb+d)\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}
Add components.
k(ab)=(kakb)k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix}
Scalar multiply each part.

Common mistakes to avoid

Adding the i component to the j component.
Add i to i and j to j separately.
Multiplying only one component by the scalar.
A scalar multiplies every component.

Key takeaways

  • Column form and i–j form are two ways to write the same vector.
  • Add/subtract by combining matching components.
  • Scalar multiplication scales every component.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.