Writing a Single Column Vector

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A column vector writes a movement as two numbers stacked vertically: the top number is the horizontal move (right positive, left negative) and the bottom number is the vertical move (up positive, down negative). In this GCSE lesson students learn how to write a single column vector from a grid, multiply by a scalar, and combine vectors with addition and subtraction. Teachers get a ready teaching sequence, an unlimited column vectors question generator, and four exam-style questions with revealable answers.

This lesson sits in the Vectors unit for GCSE and IGCSE Mathematics (Foundation and Higher) across Edexcel, AQA and OCR. At Foundation it is one of the most demanding topics on the paper; at Higher it leads into magnitude, vector paths and geometric proof. Prior knowledge is describing translations with column vectors. Next steps are magnitude of a vector, then Higher-tier vector geometry.

Browse the full sequence on the Vectors lessons hub, or follow the GCSE Foundation and Higher / IGCSE schemes of work.

Video Tutorial

Watch the worked examples for writing a single column vector, including scalar multiplication and combining vectors on a grid.

Click here to view the video on YouTube.

Lesson Overview

Differentiated Learning Objectives

• 🟡 All students should represent a column vector as a translation on a grid
• 🟢 Most students should use algebraic methods to add and subtract with column vectors
• 🟣 Some students should solve identities involving column vectors

Key Concepts

What is a column vector?

A column vector describes a translation using two components written one above the other inside brackets. The top component is the change in x (right is positive, left is negative). The bottom component is the change in y (up is positive, down is negative). Unlike a coordinate pair, a column vector is a directed movement, not a position on the grid.

\mathbf{a}=\left(\begin{matrix}5\\2\end{matrix}\right)\quad\text{means 5 right and 2 up}

How to write a column vector

• Count squares right or left for the top number
• Count squares up or down for the bottom number
• Write both numbers stacked vertically inside large brackets (no fraction line)
• To combine vectors: multiply each component by the scalar, then add or subtract component by component

Column vector formula: scalar multiplication, addition and subtraction

If

\mathbf{a}=\left(\begin{matrix}a_1\\a_2\end{matrix}\right),\quad\mathbf{b}=\left(\begin{matrix}b_1\\b_2\end{matrix}\right)

then

k\mathbf{a}=\left(\begin{matrix}ka_1\\ka_2\end{matrix}\right),\quad
\mathbf{a}+\mathbf{b}=\left(\begin{matrix}a_1+b_1\\a_2+b_2\end{matrix}\right),\quad
\mathbf{a}-\mathbf{b}=\left(\begin{matrix}a_1-b_1\\a_2-b_2\end{matrix}\right)

Worked example: write a – 2b as a single column vector

Given

\mathbf{a}=\left(\begin{matrix}5\\2\end{matrix}\right),\quad\mathbf{b}=\left(\begin{matrix}3\\-1\end{matrix}\right)

calculate a – 2b (the style of question that appears on Foundation GCSE papers):

\mathbf{a}-2\mathbf{b}
=\left(\begin{matrix}5\\2\end{matrix}\right)-2\left(\begin{matrix}3\\-1\end{matrix}\right)
=\left(\begin{matrix}5\\2\end{matrix}\right)-\left(\begin{matrix}6\\-2\end{matrix}\right)
=\left(\begin{matrix}-1\\4\end{matrix}\right)

Check the y-component carefully: 2 – (2 x -1) = 2 – (-2) = 2 + 2 = 4. Then draw the result as an arrow 1 left and 4 up. Examiners report that many students stop after the algebra and lose the drawing mark.

Column Vectors Question Generator

Generate unlimited column vector questions at three difficulty levels, without searching for another worksheet. Use this for whiteboard practice, a quick plenary, differentiated independent work or cover lessons.

• Select Level 1 to Level 3 to match your class
• Choose 4 or 8 questions, then click New Questions
• Reveal answers with Answer Next for paced feedback, or Answer All when you are ready to mark

Open the Column Vectors Question Generator in a new tab if the embed does not display on your device.

Mr Mathematics members use question generators like this across the secondary curriculum, alongside other lessons in the vectors scheme of work, ready-to-teach PowerPoints and differentiated worksheets.

Exam-Style Questions

These four questions progress from reading vectors off a grid through to combining named vectors algebraically. They cover the Foundation and Higher question types that appear most often.

• Lesson plenary: project one question, allow two to three minutes, then reveal the answer
• Starter for the next lesson: display all four for retrieval before magnitude or vector paths
• Homework: Questions 1 and 2 for standard practice; Questions 3 and 4 for Higher challenge

Question 1

Show worked solution

Part a) Draw -2a
Multiply each component of a by -2. This reverses the direction and doubles the length. Draw the arrow from any point and label it -2a.

Part b) a + 2b
Double each component of b, then add component by component. Watch negative y-components: 2 x (-3) = -6, not -3.

Question 2

Show worked solution

a) 2a = (4, 2): 4 right and 2 up
b) a + b = (0, 2): straight up 2
c) a – b = (4, 0): 4 right. Check the double negative: 2 – (-2) = 4.

Question 3

Show worked solution

3a = (9, 3), 2b = (2, -4), then 3a – 2b = (7, 7).
Key check: y-component 3 – (-4) = 7 (the double-negative trap).

Question 4

Show worked solution

Path: PR = PQ + QR. Reverse RQ to get QR = (2, -4).
Then PR = (5, 3) + (2, -4) = (7, -1).
Common error: using RQ without reversing it first.

Learning Checklist

Students should now be able to:

• ✅ Explain what a column vector is and read one from a grid
• ✅ Multiply a column vector by a positive or negative scalar
• ✅ Add and subtract column vectors component by component
• ✅ Write an expression such as 2a – 3b as a single column vector
• ✅ Draw the resulting vector as a directed arrow, not a point

Common Mistakes

• ❌ Double-negative sign slips: when subtracting a vector with a negative component, write every step. For example, 2 – (2 x -1) = 2 + 2 = 4, not 2 – 2.
• ❌ Writing a coordinate pair instead of a column vector: answers must be stacked vertically inside brackets, not written as (-1, 4).
• ❌ Adding a fraction line or omitting brackets: examiners treat a horizontal line between components as incorrect notation and may withhold the communication mark.
• ❌ Calculating correctly but plotting a point: a column vector is a directed arrow from a starting position, not a single plotted coordinate.
• ❌ Swapping the x and y components: top is horizontal, bottom is vertical. Linking back to translations helps secure this.

Frequently Asked Questions

What to Teach Next

Keep the vectors sequence coherent: secure single column vectors, then move on to magnitude and Higher-tier geometry.