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Vector notation

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Here we will learn about column vectors, including how to write a column vector and how to draw a diagram of a column vector.

There are also vector worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

A column vector is a way of writing a vector which gives information about the vector. It is split into a horizontal component and a vertical component.

There is a **horizontal** component, also known as the \textbf{x} **component** This is the **top number** in the column vector and tells us how many spaces to the right or left to move.

If the number is **positive**, the direction is to the **right**.

If the number is **negative**, the direction is to the **left**.

There is a **vertical component**, also known as the \textbf{y} **component. **This is the **bottom number** in the column vector and tells us how many spaces up or down to move.

If the number is **positive**, the direction is **upwards**.

If the number is **negative**, the direction is **downwards**.

E.g.

Vector \textbf{a} can be written as the column vector \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

\textbf{a}= \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix} \text{is} \begin{pmatrix} 3 \ \text{right}\\ 2 \ \text{up}\\ \end{pmatrix}Notice the horizontal component and the vertical component make a right-angled triangle.

E.g.

Vector \textbf{b} can be written as the column vector \begin{pmatrix} \; 3 \;\\ \; -4 \; \end{pmatrix}

\textbf{b}= \begin{pmatrix} \; 3 \;\\ \; -4 \; \end{pmatrix} \text{is} \begin{pmatrix} 3 \ \text{right}\\ -4 \ \text{down}\\ \end{pmatrix}Notice the horizontal component and the vertical component make a right-angled triangle.

In order to write a vector as a column vector:

**Work out the horizontal component (**\textbf{x}**component).****Work out the vertical component (**\textbf{y}**component).****Write the column vector.**

Get your free column vector worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free column vector worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONWrite vector \textbf{a} as a column vector.

**Work out the horizontal component (**\textbf{x}**component).**

From the starting point of the vector, draw a horizontal line.

This line is 4 squares to the right.

2**Work out the vertical component (** \textbf{y} ** component).**

From the end of the horizontal component, draw a vertical line to the end of the vector.

This line is 3 squares up.

3**Write the column vector.**

Write the horizontal component and the vertical component in a column vector.

Vector \textbf{a} as a column vector is:

\textbf{a}= \begin{pmatrix} \; 4 \;\\ \; 3 \; \end{pmatrix}Write vector \textbf{b} as a column vector.

**Work out the horizontal component (** \textbf{x} **component).**

From the starting point of the vector, draw a horizontal line.

This line is 4 squares to the right.

**Work out the vertical component (** \textbf{y} ** component).**

From the end of the horizontal component, draw a vertical line to the end of the vector.

This line is 1 square down.

**Write the column vector.**

Write the horizontal component and the vertical component in a column vector.

Vector \textbf{b} as a column vector is:

\textbf{b}=
\begin{pmatrix}
\; 4 \;\\
\; -1 \;
\end{pmatrix}

Write vector \textbf{v} as a column vector.

**Work out the horizontal component (** \textbf{x} **component).**

From the starting point of the vector, draw a horizontal line. We are trying to make a right-angled triangle.

This line is 2 squares to the left.

**Work out the vertical component (** \textbf{y} ** component).**

From the end of the horizontal component, draw a vertical line to the end of the vector.

This line is 1 square down.

**Write the column vector.**

Write the horizontal component and the vertical component in a column vector.

Vector \textbf{v} as a column vector is:

\textbf{v}=
\begin{pmatrix}
\; -2 \;\\
\; -1 \;
\end{pmatrix}

In order to draw a diagram of a column vector:

**Draw the horizontal component****(**\textbf{x}**component).**\textbf{y}**Draw the vertical component**(**component).****Draw the vector.**

Draw a diagram of the column vector \begin{pmatrix} \; 2 \;\\ \; 5 \; \end{pmatrix}

** Draw the horizontal component (** \textbf{x}

On the grid choose a starting point and draw the horizontal component

The top number is 2 so we draw a line 2 squares to the right .

** Draw the vertical component (** \textbf{y}

From the end of the horizontal component, draw the vertical component.

The bottom number is 5 so we draw a line 5 squares up.

**Draw the vector.**

Join up the starting point and the end point and remember to put the direction arrow on the line.

Draw a diagram of the column vector \begin{pmatrix} \; -3 \;\\ \; 1 \; \end{pmatrix}

** Draw the horizontal component (** \textbf{x}

On the grid choose a starting point and draw the horizontal component

The top number is -3 so we draw a line 3 squares to the left .

** Draw the vertical component (** \textbf{y}

From the end of the horizontal component, draw the vertical component.

The bottom number is 1 so we draw a line 1 square up.

**Draw the vector.**

Join up the starting point and the end point and remember to put the direction arrow on the line.

Draw a diagram of the column vector \begin{pmatrix} \; -6 \;\\ \; -1 \; \end{pmatrix}

** Draw the horizontal component (** \textbf{x}

On the grid choose a starting point and draw the horizontal component

The top number is -6 so we draw a line 6 squares to the left.

** Draw the vertical component (** \textbf{y}

From the end of the horizontal component, draw the vertical component.

The bottom number is -1 so we draw a line 1 square down.

**Draw the vector.**

Join up the starting point and the end point and remember to put the direction arrow on the line.

**Make sure the signs are correct**

Remember:

If the top number is positive, the direction is to the right.

If the top number is negative, the direction is to the left.

If the bottom number is positive, the direction is upwards.

If the bottom number is negative, the direction is downwards.

**Column vectors notation**

Column vectors only have 2 numbers within the brackets; a top number and a bottom number. There is no need for any other punctuation marks such as commas or semicolons and there is no need for a line to separate the numbers.

Column vector is part of our series of lessons to support revision on vectors. You may find it helpful to start with the main vectors lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. Write this vector as a column vector:

\begin{pmatrix}
\; -4 \;\\
\; 1 \;
\end{pmatrix}

\begin{pmatrix}
\; 4 \;\\
\; 1 \;
\end{pmatrix}

\begin{pmatrix}
\; 1 \;\\
\; 4 \;
\end{pmatrix}

\begin{pmatrix}
\; -1 \;\\
\; 4 \;
\end{pmatrix}

Draw a horizontal line and a vertical line and count the squares.

\begin{pmatrix} \; 4 \;\\ \; 1 \; \end{pmatrix}

2. Write this vector as a column vector:

\begin{pmatrix}
\; 2 \;\\
\; -4 \;
\end{pmatrix}

\begin{pmatrix}
\; 4 \;\\
\; -2 \;
\end{pmatrix}

\begin{pmatrix}
\; -4 \;\\
\; -2 \;
\end{pmatrix}

\begin{pmatrix}
\; -2 \;\\
\; 4 \;
\end{pmatrix}

Draw a horizontal line and a vertical line and count the squares.

\begin{pmatrix} \; -2 \;\\ \; 4 \; \end{pmatrix}

3. Write the vector \textbf{x} as a column vector:

\begin{pmatrix}
\; 1 \;\\
\; 3 \;
\end{pmatrix}

\begin{pmatrix}
\; -1 \;\\
\; 3 \;
\end{pmatrix}

\begin{pmatrix}
\; 1 \;\\
\; -3 \;
\end{pmatrix}

\begin{pmatrix}
\; -1 \;\\
\; -3 \;
\end{pmatrix}

Draw a horizontal line and a vertical line and count the squares.

The column vector for \textbf{x} is

\begin{pmatrix} \; 1 \;\\ \; -3 \; \end{pmatrix}

4. Draw the vector

\begin{pmatrix} \; 1 \;\\ \; 4 \; \end{pmatrix}

The top number of the column vector is 1 . This is the horizontal component. Use this to draw a horizontal line to the right. The bottom number of the column vector is 4 . This is the vertical component. Use this to draw a vertical line upwards.

5. Draw the vector

\begin{pmatrix} \; -3 \;\\ \; -2 \; \end{pmatrix}

The top number of the column vector is -3 . This is the horizontal component. Use this to draw a horizontal line to the left. The bottom number of the column vector is -2 . This is the vertical component. Use this to draw a vertical line downwards.

6. Draw the vector

\begin{pmatrix} \; -5 \;\\ \; 2 \; \end{pmatrix}

The top number of the column vector is -5 . This is the horizontal component. Use this to draw a horizontal line to the left. The bottom number of the column vector is 2 . This is the vertical component. Use this to draw a vertical line upwards.

1. Which is the correct column vector for this vector?

\begin{aligned} &\quad \text{A} \quad \quad\quad \quad \;\; \text{B} \quad \quad \quad \quad \text{C} \quad \quad \quad \quad \text{D} \\\\ &\begin{pmatrix} \; 4 \;\\ \; 0 \; \end{pmatrix} \quad \quad \begin{pmatrix} \; -4 \;\\ \; 0 \; \end{pmatrix} \quad \quad \begin{pmatrix} \; 0 \;\\ \; 4 \; \end{pmatrix} \quad \quad \begin{pmatrix} \; 0 \;\\ \; -4 \; \end{pmatrix} \end{aligned}

**(1 mark)**

Show answer

\begin{aligned}
&\quad \text{B} \\\\
&\begin{pmatrix}
\; -4 \;\\
\; 0 \;
\end{pmatrix}
\end{aligned}

**(1)**

2. The column vector \begin{pmatrix} \; 4 \;\\ \; a \; \end{pmatrix} represents:

What is the value of a ?

**(1 mark)**

Show answer

a=-3

**(1)**

3. Write the column vector for this vector

**(2 marks)**

Show answer

\begin{pmatrix}
\; 2 \;\\
\; -5 \;
\end{pmatrix}

(For the correct horizontal component)

**(1)**

(For the correct vertical component)

**(1)**

You have now learned how to:

- How to write a vector as a column vector
- How to draw a diagram of a column vector

**Not** covered in GCSE: we can transpose a column vector to write it as a row vector (and vice versa). These look like co-ordinates, but do not have commas.

Vectors can also be extended into A Level Maths and Further Maths by learning how to multiply two vectors together using the dot product.

Column vectors are a simple example of matrices. In GCSE maths we have a single column. Matrices are studied in A Level Further Maths. The number of columns and rows will be more than 1 . Matrix multiplication can be studied along with finding the inverse of a matrix. We can also find the determinant of a matrix and go further and look into eigenvalues and eigenvectors.

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