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Vectors Revision


A vector is something with both magnitude and direction. On diagrams they are denoted by an arrow, where the length tells us the magnitude and the arrow tells us direction.

  • You will need to add and subtract vectors
  • You will also need to multiply vectors and understand scalar multiples of vectors 
Vector Direction Arrow

Make sure you are happy with the following topics before continuing:

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Adding and subtracting vectors Diagram

Adding and Subtracting Vectors

When we add vectors:


this takes you from the start of \mathbf{a} to the end of \mathbf{b} (right).

The negative of a vector has the same magnitude of the original vector, it just goes in the exact opposite direction.

When we subtract vectors:


we add on the negative of the vector that is being subtracted.

Adding and subtracting vectors Diagram
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Multiplying Vectors

Multiplying vectors by a Scalar – We can  multiply a vector by a number. For example, we can multiply \mathbf{a} by 3:


This means the the vectors are added end to end.

We can multiply more complicated vectors 


Note: all vectors here are written in bold. When you’re writing this by hand, you should underline each letter that represents a vector.

Scalar multiples – Scalar multiples are all parallel to each other:

\mathbf{3\textcolor{red}{a}}+\mathbf{3\textcolor{blue}{b}} is parallel to \mathbf{\textcolor{red}{a}}+\mathbf{\textcolor{blue}{b}}


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Vector Notation Diagram

Vector Notation

A vector from point X to point Y will be shown as \overrightarrow{XY}.

Vector Notation Diagram

So from the diagram shown we can see that:

\overrightarrow{XY} = \mathbf{a}

\overrightarrow{YZ} = \mathbf{b}

\overrightarrow{ZX} = \mathbf{c}

If we go against the arrow the vector becomes negative.

\overrightarrow{YX} = \mathbf{-a}

We can also combine vectors, meaning

\overrightarrow{XZ} = \mathbf{a + b} = \mathbf{-c}

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Example: Vectors and Ratios

In the diagram below, we have vectors

\overrightarrow{AB}=3\mathbf{a} and \overrightarrow{AC}=4\mathbf{b} .

Point D lies on the  line BC such that BD:DC=1:3.

Write the vector \overrightarrow{AD} in terms of \mathbf{a} and \mathbf{b}.

[3 marks]

Involving ratios in Vectors

To find \overrightarrow{AD}, we are going to add the vectors


We know \overrightarrow{AB}=3\mathbf{a} but we need to find \overrightarrow{BD}

We know BD:DC=1:3, so the point D is \dfrac{1}{4} of the way along \overrightarrow{BC}


Now we need to multiply this by \dfrac{1}{4}


Now we need to add  \overrightarrow{AB} to the answer:


Thus, we have our answer in terms of \mathbf{a} and \mathbf{b}.

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Vectors Example Questions

First we can find the vector \overrightarrow{AC}


\overrightarrow{AC} =\overrightarrow{AB} +\overrightarrow{BC} =3\mathbf{a} +2\mathbf{b}


Since M is the midpoint of AC we know that \overrightarrow{AM}=\dfrac{1}{2}\overrightarrow{AC}  


\overrightarrow{AM}=\dfrac{1}{2}(3\mathbf{a} +2\mathbf{b})  = \dfrac{3}{2}\mathbf{a} +\mathbf{b}

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The vector \overrightarrow{EF} is the same as:




First, we can find the vector \overrightarrow{ED} as \dfrac{1}{2}\mathbf{b} .


Then we also know \overrightarrow{DC} = \overrightarrow{AB}, hence \overrightarrow{DC}=2\mathbf{a} .


The final section \overrightarrow{CF} = \dfrac{3}{5}\times 2\mathbf{a}=\dfrac{6}{5}\mathbf{a}




\overrightarrow{EF}= \dfrac{1}{2} \mathbf{b}+2\mathbf{a}+\dfrac{6}{5}\mathbf{a}=3.2\mathbf{a}+ \dfrac{1}{2} \mathbf{b}

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First we can find the vector \overrightarrow{BC}


\overrightarrow{BC} =-\overrightarrow{AB} +\overrightarrow{AC} =-\mathbf{a} +2\mathbf{b}


Then we can find:


\overrightarrow{BD} =\dfrac{3}{5}\overrightarrow{BC}  =-\dfrac{3}{5}\mathbf{a} +\dfrac{6}{5}\mathbf{b}




\begin{aligned} \overrightarrow{AD} &=\overrightarrow{AB}+\overrightarrow{BD}  \\ &=\mathbf{a} +\bigg(-\dfrac{3}{5}\mathbf{a} +\dfrac{6}{5}\mathbf{b}\bigg) \\ & =\dfrac{2}{5}\mathbf{a}+\dfrac{6}{5}\mathbf{b} \end{aligned}




\begin{aligned} \overrightarrow{AE} &=\dfrac{1}{3}\overrightarrow{AD} \\ &=\dfrac{1}{3}\bigg(\dfrac{2}{5}\mathbf{a}+\dfrac{6}{5}\mathbf{b}\bigg) \\ &= \dfrac{2}{15}\mathbf{a}+\dfrac{2}{5}\mathbf{b} \end{aligned}

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We will find \overrightarrow{AC} by doing,






\overrightarrow{AC} =\mathbf{b}-\mathbf{a}+\mathbf{b}= 2\mathbf{b}-\mathbf{a}

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We will find \overrightarrow{EB} by doing




The first vector is straightforward, because we know \overrightarrow{AE}, and that is just the same vector in the opposite direction. So, we get:




Now we need \overrightarrow{AB}. Since B is the midpoint of AC (given in the question), we must have that \overrightarrow{AB}=\frac{1}{2} \overrightarrow{AC}. Therefore, looking at the diagram, we get that:




We’re given the second part of this, \overrightarrow{DC}=2\mathbf{a}+4\mathbf{b}, and since E is the midpoint of AD, we can also work out the first part:




Now, we have everything we need and can go back through our work, filling in the gaps we find:


\overrightarrow{AB}=\dfrac{1}{2}\left(6\mathbf{a}-4\mathbf{b}+2\mathbf{a} +4\mathbf{b}\right)=\dfrac{1}{2}\left(8\mathbf{a}\right)=4\mathbf{a}


Finally we have that:




If \overrightarrow{EB} and \overrightarrow{DC} are parallel, then one must be a multiple of the other. Well, if we multiply \overrightarrow{EB} by 2 then,




We have shown that 2\overrightarrow{EB}=\overrightarrow{DC}, therefore, the two lines must be parallel.

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Vectors Worksheet and Example Questions

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