Vectors

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Tutor: Bilal

Summary

​​​Vectors

In a nutshell

You need to be able to recall the definition of a vector as well as how to add, subtract, and multiply with them.


Definitions

A vector is a quantity with both magnitude and direction. For example, "5 metres north" is a vector.

A scalar is a quantity with magnitude only. For example, "5 metres" is a scalar as you don't know in which direction.



Representing vectors

There are different ways to represent vectors:


Column vectors

Vectors are commonly represented in column form. Column vectors are written like this: (ab)\begin{pmatrix}a\\b\end{pmatrix}. This means move aa​​ steps right and bb steps up.

Bold lowercase letters

Bold lowercase letters are used as a quick way to refer to a vector. For example, you may see a\textbf{a} = (11)\begin{pmatrix}1\\1\end{pmatrix}​.

Uppercase letters with an arrow

The vector from the point A to the point B can be denoted as AB\overrightarrow{AB}​​.​

On a graph

On a graph, draw vectors with an arrow to point in the direction it is moving in.



Adding and subtracting vectors

To add and subtract column vectors, add and subtract each row individually.


Example 1


If a=(21)\textbf{a}=\begin{pmatrix}2\\1\end{pmatrix} and b=(63)\textbf{b}=\begin{pmatrix}-6\\3\end{pmatrix}, work out a+b\textbf{a}+\textbf{b} and ab\textbf{a}-\textbf{b}:

a+b=(21)+(63)=(2+(6)1+3)=(44)\textbf{a}+\textbf{b}=\begin{pmatrix}2\\1\end{pmatrix}+\begin{pmatrix}-6\\3\end{pmatrix}=\begin{pmatrix}2+(-6)\\1+3\end{pmatrix}=\begin{pmatrix}-4\\4\end{pmatrix}​​


a+b=(44)\underline{\textbf{a}+\textbf{b}=\begin{pmatrix}-4\\4\end{pmatrix}}​​


ab=(21)(63)=(2(6)13)=(82)\textbf{a}-\textbf{b}=\begin{pmatrix}2\\1\end{pmatrix}-\begin{pmatrix}-6\\3\end{pmatrix}=\begin{pmatrix}2-(-6)\\1-3\end{pmatrix}=\begin{pmatrix}8\\-2\end{pmatrix}​​


ab=(82)\underline{\textbf{a}-\textbf{b}=\begin{pmatrix}8\\-2\end{pmatrix}}​​



Multiplying a vector with a scalar

To multiply a vector with a scalar, multiply each number in the vector by the scalar.


Example 2


If a=(41)\textbf{a}=\begin{pmatrix}4\\-1\end{pmatrix}, work out 3a3\textbf{a}, and 2a-2\textbf{a}:

3a=3(41)=(3×43×1)=(123)3\textbf{a}=3\begin{pmatrix}4\\-1\end{pmatrix}=\begin{pmatrix}3\times4\\3\times-1\end{pmatrix}=\begin{pmatrix}12\\-3\end{pmatrix}


3a=(123)\underline{3\textbf{a}=\begin{pmatrix}12\\-3\end{pmatrix}}​​​


2a=2(41)=(2×42×1)=(82)-2\textbf{a}=-2\begin{pmatrix}4\\-1\end{pmatrix}=\begin{pmatrix}-2\times4\\-2\times-1\end{pmatrix}=\begin{pmatrix}-8\\2\end{pmatrix}​​​


2a=(84)\underline{-2\textbf{a}=\begin{pmatrix}-8\\4\end{pmatrix}}​​



Vector geometry: arithmetic

Adding, subtracting, and multiplying vectors each have their own geometric interpretations.

Addition

Adding two vectors a\textbf{a}​ and b\textbf{b}​ gives the vector a+b\textbf{a}+\textbf{b}​, which represents travelling along the vector a\textbf{a}​ then along the vector b\textbf{b} in one journey.

Maths; Trigonometry; KS4 Year 10; Vectors

Multiplication by a positive scalar

Multiplying a vector a\textbf{a}​ by a positive scalar (number) kk gives the vector kak\textbf{a}​, which is a vector in the same direction as a\textbf{a}​, but its length is multiplied by a factor of kk​.

Maths; Trigonometry; KS4 Year 10; Vectors

Negative of a vector/multiplication by 1-1

The negative of a vector a\textbf{a}​ is denoted as (a-\textbf{a}), which means it is the same length as a\textbf{a}​, but going in the opposite direction.

Maths; Trigonometry; KS4 Year 10; Vectors

Subtraction

ab=a+(b)\textbf{a}-\textbf{b}=\textbf{a}+(-\textbf{b})​. So subtracting the vector b\textbf{b}​ from a\textbf{a}​ represents going along the vector a\textbf{a}​, and then going along the vector b-\textbf{b}​.

Maths; Trigonometry; KS4 Year 10; Vectors


Example 3

In the diagram, find the vector AB\overrightarrow{AB}:


Maths; Trigonometry; KS4 Year 10; Vectors


To get from the point AA to the point BB​, you go from AA​ to OO​, then from OO​ to BB​. Mathematically:

AB=AO+OB\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}​​


From the diagram, OB=b\overrightarrow{OB}=\textbf{b}.

If OA=a\overrightarrow{OA}=\textbf{a}, then AO=a\overrightarrow{AO}=-\textbf{a} because it is going in the opposite direction.

Therefore:

 AB=AO+OB=(a)+b=ba\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}=(-\textbf{a})+\textbf{b}=\textbf{b}-\textbf{a}.


AB=ba\underline{\overrightarrow{AB}=\textbf{b}-\textbf{a}}​​




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Exercises

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FAQs - Frequently Asked Questions

How do you add two vectors together?

How do you read a column vector?

What is a vector?

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