Introduction to Column Vectors

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Basic Vectors

Basic vectors

A column vector can be used to describe how to get from one point to another point
- This is also called a translation vector
- $(\begin{matrix} 6 \\ 3 \end{matrix})$ means 6 units to the right and 3 units up

Adding and subtracting vectors is done by looking at the top numbers and bottom numbers separately
To add column vectors
- Add the top numbers together
- Add the bottom numbers together
  - $(\begin{matrix} 5 \\ 2 \end{matrix}) + (\begin{matrix} 3 \\ - 1 \end{matrix}) = (\begin{matrix} 5 + 3 \\ 2 + (- 1) \end{matrix}) = (\begin{matrix} 8 \\ 1 \end{matrix})$
To subtract column vectors
- Subtract the second top number from the first
- Subtract the second bottom number from the first
  - $(\begin{matrix} 5 \\ 2 \end{matrix}) - (\begin{matrix} 3 \\ - 1 \end{matrix}) = (\begin{matrix} 5 - 3 \\ 2 - (- 1) \end{matrix}) = (\begin{matrix} 2 \\ 3 \end{matrix})$

A scalar is a number not a vector
- It does not have a direction
To multiply a column vector by a scalar
- Multiply the top number by the scalar
- Multiply the bottom number by the scalar
  - $3 (\begin{matrix} 2 \\ - 1 \end{matrix}) = (\begin{matrix} 3 \times 2 \\ 3 \times (- 1) \end{matrix}) = (\begin{matrix} 6 \\ - 3 \end{matrix})$

You need to follow the order of operations
- $2 (\begin{matrix} 5 \\ 2 \end{matrix}) + 5 (\begin{matrix} 3 \\ - 1 \end{matrix})$
STEP 1
Multiply each vector by the scalar in front of it
- $(\begin{matrix} 2 \times 5 \\ 2 \times 2 \end{matrix}) + (\begin{matrix} 5 \times 3 \\ 5 \times (- 1) \end{matrix}) = (\begin{matrix} 10 \\ 4 \end{matrix}) + (\begin{matrix} 15 \\ - 5 \end{matrix})$
STEP 2
Add or subtract the new column vectors
- $(\begin{matrix} 10 + 15 \\ 4 + (- 5) \end{matrix}) = (\begin{matrix} 25 \\ - 1 \end{matrix})$

Worked example

$a = (\begin{matrix} p \\ 3 \end{matrix})$ and $b = (\begin{matrix} - 2 \\ 1 \end{matrix})$ .

Given that $2 a + 3 b = (\begin{matrix} 4 \\ q \end{matrix})$ , find the value of $p$ and the value of $q$ .

Answer:

Write the left-side side as one vector
Multiply each vector by the scalar in front of it

$(\begin{matrix} 2 p \\ 6 \end{matrix}) + (\begin{matrix} - 6 \\ 3 \end{matrix}) = (\begin{matrix} 4 \\ q \end{matrix})$

Add the vectors together

$(\begin{matrix} 2 p - 6 \\ 9 \end{matrix}) = (\begin{matrix} 4 \\ q \end{matrix})$

The top components are equal
Form and solve an equation

$\begin{array}{rcl} 2 p - 6 & = & 4 \\ 2 p & = & 10 \\ p & = & 5 \end{array}$

The bottom components are equal

$9 = q$

$p = 5$ and $q = 9$