Vectors in Two Dimensions

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Describe a translation using a vector represented by (x/y), AB or a
  2. Add and subtract vectors
  3. Multiply a vector by a scalar
  4. Understand that vectors will be printed as AB or a

What is a Vector?

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A vector is a quantity that has both magnitude (size) and direction. Think of it as an instruction for movement: how far to go and in which direction.

For example, if you walk 5 steps north, that's a vector - it tells you the distance (5 steps) and the direction (north).

Vectors are different from ordinary numbers (called scalars), which only have size. For example, "5" is just a number, but "5 steps north" is a vector.

How Vectors are Written

Vectors can be written in three different ways:

  1. Column notation: (xy)\begin{pmatrix} x \\ y \end{pmatrix} where x is the horizontal movement and y is the vertical movement

  2. Using two points: AB (printed in bold or with an arrow above: AB→\overrightarrow{AB}), which means the vector from point A to point B

  3. Using a single letter: a (printed in bold or with an underline: a‾\underline{a}), just a name for the vector

Example: The vector (32)\begin{pmatrix} 3 \\ 2 \end{pmatrix} means "move 3 units to the right and 2 units up."

Understanding Column Vectors

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A vector written as (xy)\begin{pmatrix} x \\ y \end{pmatrix} is called a column vector.

  • The top number (x) tells you the horizontal movement:

    • Positive means move right
    • Negative means move left

  • The bottom number (y) tells you the vertical movement:

    • Positive means move up
    • Negative means move down

Examples of Column Vectors

Example 1: (43)\begin{pmatrix} 4 \\ 3 \end{pmatrix}

  • Move 4 units right
  • Move 3 units up

Example 2: (−25)\begin{pmatrix} -2 \\ 5 \end{pmatrix}

  • Move 2 units left (because of the negative sign)
  • Move 5 units up

Example 3: (3−4)\begin{pmatrix} 3 \\ -4 \end{pmatrix}

  • Move 3 units right
  • Move 4 units down (because of the negative sign)

Example 4: (−1−6)\begin{pmatrix} -1 \\ -6 \end{pmatrix}

  • Move 1 unit left
  • Move 6 units down

Describing Translations Using Vectors

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A translation is a transformation that slides a shape from one position to another without rotating or flipping it. Every point on the shape moves the same distance in the same direction.

We use vectors to describe translations precisely.

How to Describe a Translation

If a point moves from position A to position B, the translation vector is written as AB or (xy)\begin{pmatrix} x \\ y \end{pmatrix}.

Example: If point P at coordinates (2, 3) moves to point Q at coordinates (5, 7), what is the translation vector PQ?

Solution:

  • Horizontal movement: 5 - 2 = 3 (move 3 right)
  • Vertical movement: 7 - 3 = 4 (move 4 up)
  • Translation vector PQ = (34)\begin{pmatrix} 3 \\ 4 \end{pmatrix}

Another Way to Think About It

To find a vector from point A(x₁, y₁) to point B(x₂, y₂):

Vector AB→=(x2−x1y2−y1)\text{Vector } \overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}

Example: Find the vector from A(1, 4) to B(6, 2).

Solution: AB→=(6−12−4)=(5−2)\overrightarrow{AB} = \begin{pmatrix} 6 - 1 \\ 2 - 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}

This means move 5 units right and 2 units down.

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