13.1 Vector Notation

2026 📋 Syllabus Objectives

By the end of these notes, you will be able to:

  • Understand what a vector is and how it differs from a scalar
  • Recognise and use all forms of vector notation: column vectors, directed line segments, bold letters, and unit vector form
  • Write vectors correctly using the right notation in any given situation

📖 Main Content

https://cambridgeassistant.com

What is a Vector?

A vector is a quantity that has both a size (called its magnitude) and a direction.

Think of it this way: if someone tells you to "walk 5 km", that is just a size — no direction given. But if they say "walk 5 km north", that is a vector — it has both size and direction.

Compare this with a scalar, which is a quantity that has only size and no direction. Temperature, time, and mass are all scalars.

Example: Speed is a scalar (just a size). Velocity is a vector (size + direction). A car travelling at 60 km/h is scalar; a car travelling at 60 km/h due east is a vector.

The Four Forms of Vector Notation

Vectors can be written in four different ways. You must be able to recognise and use all of them correctly. The Cambridge O Level exam may give you a vector in any of these forms.

✏️ Form 1 — Column Vector

A column vector writes the vector as two numbers stacked on top of each other inside brackets:

(ab)\begin{pmatrix} a \\ b \end{pmatrix}

  • The top number (a) tells you the horizontal movement — how far the vector moves left or right.
    • Positive a → moves to the right
    • Negative a → moves to the left
  • The bottom number (b) tells you the vertical movement — how far the vector moves up or down.
    • Positive b → moves up
    • Negative b → moves down

Example: (34)\begin{pmatrix} 3 \\ 4 \end{pmatrix} This vector moves 3 units to the right and 4 units up.

Example: (−25)\begin{pmatrix} -2 \\ 5 \end{pmatrix} This vector moves 2 units to the left and 5 units up.

Example: (6−3)\begin{pmatrix} 6 \\ -3 \end{pmatrix} This vector moves 6 units to the right and 3 units down.

✏️ Form 2 — Directed Line Segment Notation: AB→\overrightarrow{AB}

This notation uses two capital letters with an arrow on top. It represents the vector that starts at point A and ends (points to) point B.

  • The first letter is the starting point (the tail of the arrow).
  • The second letter is the ending point (the tip of the arrow).
  • The arrow on top always points from the first letter to the second letter.

Example: AB→\overrightarrow{AB} means "the vector that starts at A and travels to B."

⚠️ Important: Direction matters! AB→\overrightarrow{AB} and BA→\overrightarrow{BA} are not the same vector — they point in opposite directions.

BA→=−AB→\overrightarrow{BA} = -\overrightarrow{AB}

This means BA→\overrightarrow{BA} has the same size as AB→\overrightarrow{AB}, but travels in the opposite direction.

Example in geometry: In a triangle with vertices A, B, and C:

  • AB→\overrightarrow{AB} goes from A to B
  • BC→\overrightarrow{BC} goes from B to C
  • CA→\overrightarrow{CA} goes from C back to A

Sign in to view full notes