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 Video by Zeeshan Zamurred Vectors

Edexcel AS Level Maths: 11.2 Representing Vectors (Column / i, j)Watch the full walkthrough before the notes below.

Open on YouTube

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

Two notations

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

(32) = 3 i + 2 j

Top number =

i

component; bottom =

j

component.

Adding in components

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

(a b) + (c d) = (a + c b + d)

Combine like components.

1Add

i

parts:

2 + 4 = 6

2Add

j

parts:

3 - 1 = 2

Answer

6 i + 2 j

Scalar multiplication

To multiply by a scalar, multiply $each$ component.

1Multiply each component by 3.

Answer

(6 - 3)

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

Formula recap

(a b) = a i + b j

Column ↔ i–j form.

(a b) + (c d) = (a + c b + d)

Add components.

k (a b) = (k a k b)

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

Ready to lock in Representing Vectors? Pick a mode and earn XP & Dobloons.