How To Write Vector Notation

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Vector Notation

Hither we will learn nigh vector notation, including what vectors are and how we use note in mathematics to write nigh them.

At that place are also vector worksheets based on Edexcel, AQA and OCR exam questions, along with farther guidance on where to go next if you lot're nevertheless stuck.

What is vector annotation?

Vector annotation is how we write vectors in mathematics.

A vector is a quantity which has both magnitude and management. It can be used to testify a movement.

We tin can write vectors in several ways:

• Using an pointer,
• Using boldface
• Underlined.

Eastward.one thousand.

\overrightarrow{AB} =\textbf{a}=\underline{a}

This diagram shows a vector representing the move from point A to signal B.

• Parallel vectors with the same direction and the aforementioned length are the same (equivalent):

East.g.

Each of these all represent the aforementioned vector \textbf{a} .

• Parallel vectors of the aforementioned length with opposite directions are chosen negative vectors:

E.g.

A vector representing the movement from point B to point A would be in the opposite direction, simply accept the same length.

\overrightarrow{BA}=-\textbf{a}=-\underline{a}

• A vector can exist multiplied by a scalar to change the length of it:

The length is also referred to every bit the magnitude of the vector

E.thou.

• Vectors can also be added together:

E.yard.

This diagram shows how to go from point A to point C, going via point B.

Vector \textbf{a} is added to vector \textbf{b} .

\begin{aligned} \overrightarrow{Air-conditioning}&=\overrightarrow{AB}+\overrightarrow{BC}\\\\ &= \ \textbf{a} \ +\ \textbf{b}\\\\ \text{or handwritten as}\\\\ &=\ \underline{a} \ +\ \underline{b} \end{aligned}

Adding a negative vector becomes a subtraction.

\brainstorm{aligned} &\textbf{a} \ + \ - \ \textbf{b}=\textbf{a} \ - \ \textbf{b}\\\\ &\text{or handwritten as}\\\\ &\underline{a} \ + \ - \ \underline{b} = \underline{a} \ - \ \underline{b} \end{aligned}

What is vector note?

How to use vector notation

In order to employ vector notation:

1. Check the starting point and the end point.
2. Decide the route.
3. Write the vector.
4. Simplify your answer.

How to utilise vector notation

Vector notation worksheet

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Vector annotation examples

Instance 1: using vector notation

Write the vector \overrightarrow{AO} in terms of \textbf{a} and \textbf{b}

1. Bank check the starting point and the end indicate.

\overrightarrow{AO}

The vector starts at betoken A and ends at point O.

2 Make up one's mind the route.

Start at the starting point and become forth the sides of the shape to the end point.

Yet, yous can only keep the lines which take vectors.

Hither nosotros travel in the opposite management to vector \textbf{a} .

3 Write the vector.

Write the route from point A to point O.

\overrightarrow{AO}=-\textbf{a}=-\underline{a}

Example two: using vector notation

Write the vector \overrightarrow{Air-conditioning} in terms of \textbf{a} and \textbf{b}

Check the starting point and the finish point.

\overrightarrow{Air conditioning}

The vector starts at betoken A and ends at point C.

Start at the starting indicate and go along the sides of the shape to the end bespeak.

Still, y'all can only go along the lines which have vectors.

Write the route from point A to point C.

\brainstorm{aligned} \overrightarrow{Air conditioning}&=\overrightarrow{AB}+\overrightarrow{BC}\\\\ &=\ \textbf{a} \ +\ \textbf{b}\\\\ \text{or handwritten as}\\\\ &=\ \underline{a} \ +\ \underline{b} \end{aligned}

Case 3: using vector note

Write the vector \overrightarrow{BA} in terms of \textbf{a} and \textbf{b}

Bank check the starting point and the end signal.

\overrightarrow{BA}

The vector starts at signal B and ends at point A.

Start at the starting point and go forth the sides of the shape to the stop point.

However, you lot can just go along the lines which have vectors. We can non get directly from point B to bespeak A considering there is no vector. We have to become via betoken C. As we need to go backwards along vector \textbf{b} , we demand a negative vector.

Write the route from point B to point A.

\begin{aligned} \overrightarrow{BA}&=\overrightarrow{BO}+\overrightarrow{OA}\\\\ &=\ -\textbf{b} \ +\ \textbf{a}\\ \text{or handwritten as}\\\\ &=\ -\underline{b} \ +\ \underline{a} \stop{aligned}

Alternatively, the last answer could be written every bit:

\overrightarrow{BA}=\textbf{a}-\textbf{b}=\underline{a}-\underline{b}

Example 4: leaving the answer in its simplest form

Write the vector \overrightarrow{BC} in terms of \textbf{a} and \textbf{b}

Check the starting betoken and the end betoken.

\overrightarrow{BC}

The vector starts at indicate B and ends at point C.

Start at the starting point and proceed the sides of the shape to the end point.

Notwithstanding, y'all can only go along the lines which have vectors.  We tin not go directly from indicate B to signal C.  We have to go via point O and point A.  As we demand to go backwards along vector \textbf{b} , we need a negative vector.

Write the road from point B to point C.

\begin{aligned} \overrightarrow{BC}&=\overrightarrow{BO}+\overrightarrow{OA}+\overrightarrow{AC}\\\\ &=\ -\textbf{b} \ +\ \textbf{a} \ + 2\textbf{b}\\\\ \text{or handwritten as}\\\\ &=\ -\underline{b} \ +\ \underline{a} \ + \ 2\underline{b} \finish{aligned}

To simplify we collect like terms.  So the final respond is:

\overrightarrow{BC}=\textbf{a}+\textbf{b}=\underline{a}+\underline{b}

Example 5: leaving the answer in its simplest grade

Write the vector \overrightarrow{CA} in terms of \textbf{a} and \textbf{b}

Check the starting point and the terminate point.

\overrightarrow{CA}

The vector starts at indicate C and ends at point A.

Start at the starting point and become forth the sides of the shape to the finish bespeak.

However, you can only proceed the lines which have vectors. We can not go straight from bespeak C to signal A. We accept to go via betoken B and point O. As we are going in the contrary direction at times, nosotros will need negative vectors.

Write the route from bespeak C to point A.

\brainstorm{aligned} \overrightarrow{CA}&=\overrightarrow{CB}+\overrightarrow{BO}+\overrightarrow{OA}\\\\ &=\ -5\textbf{a} \ -\ ii\textbf{b} \ +\ 4\textbf{a}\\\\ \text{or handwritten as}\\\\ &=\ -5\underline{a} \ -\ 2\underline{b} \ + \ four\underline{a} \stop{aligned}

To simplify we collect similar terms.  And then the final answer is:

\overrightarrow{CA}=-\textbf{a}-two\textbf{b}=-\underline{a}-2\underline{b}

Example 6: leaving the answer in its simplest form

Here is a hexagon.

Side OA is parallel to side Be

Side OB is parallel to side CD

Side AC is parallel to side ED

Write the vector \overrightarrow{OE} in terms of \textbf{a}, \; \textbf{b} and \textbf{c}

Check the starting betoken and the terminate point.

\overrightarrow{OE}

The vector starts at betoken O and ends at indicate E.

Start at the starting point and go forth the sides of the shape to the stop point.

Nevertheless, you can but proceed the lines which have vectors. But we can use the information to add more vectors to the diagram.

Write the route from point O to indicate E.

\begin{aligned} \overrightarrow{OE}&=\overrightarrow{OA}+\overrightarrow{AC}+\overrightarrow{CD}+\overrightarrow{DE}\\\\ &=\ \ \textbf{a} \ +\ \textbf{c} \ +\ \textbf{b} \ -\ \textbf{c}\\\\ \text{or handwritten equally}\\\\ &=\ \ \underline{a} \ +\ \underline{c} \ + \ \underline{b} \ - \ \underline{c} \finish{aligned}

To simplify we collect like terms.  So the concluding answer is:

\overrightarrow{OE}=\textbf{b}+\textbf{a}=\underline{b}+\underline{a}

Alternatively, we could accept gone from bespeak O to bespeak B so to signal E.

This would be

\begin{aligned} \overrightarrow{OE}&=\overrightarrow{OB}+\overrightarrow{Exist}\\\\ &=\ \textbf{b} \ +\ \textbf{a}\\\\ \text{or handwritten as}\\\\ &=\ \underline{b} \ +\ \underline{a} \terminate{aligned}

Which is identical to our previous answer.

Common misconceptions

• Last answers may just involve one vector

A question may ask you to write a vector in terms of vector \textbf{a} and vector \textbf{b} , but the last answer may but involve i of the vectors.

• Vectors can involve fractions or decimals

Here is a diagram of vector \textbf{c} . A vector in the aforementioned direction, but half of its length will be \frac{i}{2}\textbf{c} or 0.v\textbf{a} .

Practice vector notation questions

We demand to become in the contrary direction to vector \textbf{b} , so nosotros need a negative vector \textbf{b} .

\overrightarrow{BO}=-\textbf{b}=-\underline{b}

We need to get from point O to point B via point A.

\begin{aligned} \overrightarrow{OB}&=\overrightarrow{OA}+\overrightarrow{AB}\\\\ &=\ \ \textbf{a} \ +\ iii\textbf{b}\\\\ \text{or handwritten as}\\\\ &=\ \ \underline{a} \ +\ 3\underline{b} \end{aligned}

Nosotros demand to go from point A to point B via bespeak O. We need to go in the opposite direction to vector \textbf{a} , so nosotros need a negative vector \textbf{a} .

\begin{aligned} \overrightarrow{AB}&=\overrightarrow{AO}+\overrightarrow{OB}\\\\ &= -\textbf{a} \ +\ \textbf{b}\\\\ &= \ \ \textbf{b} \ – \ \textbf{a}\\\\ \text{or handwritten as}\\\\ &=\ \ \underline{b} \ -\ \underline{a} \end{aligned}

We need to go from indicate B to point C via point A and point D. We demand to go in the opposite direction to vector 4\textbf{b} , so we need a negative vector.

We need to go in the opposite management to vector \textbf{a} , so we need some other negative vector. When we take worked out the route, nosotros need to simplify the respond.

\brainstorm{aligned} \overrightarrow{BC}&=\overrightarrow{BA} \ +\ \overrightarrow{AD} \ + \ \overrightarrow{DC}\\\\ &= -4\textbf{b} \ -\ \textbf{a} \ + \ 3\textbf{b}\\\\ &= \ \ -\textbf{b} \ + \ \textbf{a}\\\\ \text{or handwritten as}\\\\ &=\ \ -\underline{b} \ -\ \underline{a} \end{aligned}

We need to go from point B to point A via points C and D. Nosotros demand to go in the opposite management to vector 7\textbf{a} , so we need a negative vector. When nosotros have worked out the road, we demand to simplify the respond.

\brainstorm{aligned} \overrightarrow{BA}&=\overrightarrow{BC} \ + \ \overrightarrow{CD} \ + \ \overrightarrow{DA}\\\\ &= -7\textbf{a} \ +\ 3\textbf{b} \ + \ v\textbf{a}\\\\ &= \ \ -2\textbf{a} \ + \ 3\textbf{b}\\\\ \text{or handwritten every bit}\\\\ &=\ \ -ii\underline{a} \ + \ iii\underline{b} \end{aligned}

\textbf{a}+\textbf{b}+\textbf{c}

ii\textbf{a}+\textbf{b}+\textbf{c}

We need to go from indicate O to point D.  Use the facts about parallel sides to add in more than vectors.

\begin{aligned} \overrightarrow{OD}&=\overrightarrow{OE} + \overrightarrow{ED}\\\\ &=\ \textbf{c} \ + \ \textbf{b}\\\\ &= \ \textbf{b} \ + \ \textbf{c}\\\\ \text{or handwritten as}\\\\ &=\ \underline{b} \ + \ \underline{c} \stop{aligned}

Vector notation GCSE questions

1. OABC is a trapezium

AB is parallel to OC .

(a) Observe, in terms of \textbf{b} , the vector \overrightarrow{BA}

(b) Find, in terms of \textbf{a} and \textbf{b} , the vector \overrightarrow{CA}

(two marks)

Show answer

(a)

\overrightarrow{BA}=-five\textbf{b}=-v\underline{b}

(1)

(b)

\overrightarrow{CA}=\textbf{a} \ – \ 3\textbf{b}=\underline{a} \ – \ three\underline{b}

(i)

2. ABCD is a parallelogram.

The diagonals of the parallelogram intersect at O .

\begin{aligned} \overrightarrow{OA}=\textbf{a}\\ \overrightarrow{OB}=\textbf{b} \end{aligned}

(a) Find, in terms of \textbf{a} , the vector \overrightarrow{AO}

(b) Discover, in terms of \textbf{a} , the vector \overrightarrow{CA}

(c) Find, in terms of \textbf{a} and \textbf{b} , the vector \overrightarrow{BA}

(3 marks)

Evidence answer

(a)

\overrightarrow{AO}=-\textbf{a} =-\underline{a}

(1)

(b)

\overrightarrow{CA}=2\textbf{a} =2\underline{a}

(ane)

(c)

\overrightarrow{BA}=\textbf{a} \ – \ \textbf{b} =\underline{a} \ – \ \underline{b}

(one)

3. ABCD is a trapezium.

AB is parallel to DC .

(a) Discover, in terms of \textbf{a} , the vector \overrightarrow{AO}

(b) Observe, in terms of \textbf{a} and \textbf{b} , the vector \overrightarrow{DB}

(c) Find, in terms of \textbf{a} and \textbf{b} , the vector \overrightarrow{CB}

(4 marks)

Show answer

(a)

\overrightarrow{AD}=-v\textbf{a} =-5\underline{a}

(i)

(b)

\overrightarrow{DB}=v\textbf{a} \ + \ 9\textbf{b} =5\underline{a} \ + \ 9\underline{b}

(1)

(c)

\overrightarrow{CB}=-7\textbf{b} \ + \ 5\textbf{a} \ + \ 9\textbf{b} =-7\underline{b} \ + \ 5\underline{a} \ + \ 9\underline{b}

For the correct route

(1)

\overrightarrow{CB}=5\textbf{a} \ + \ two\textbf{b} = 5\underline{a} \ + \ 2\underline{b}

For the right simplified answer

(1)

Learning checklist

You have now learned how to:

• Use vector notation to write vector
• Employ vector notation to solve a geometry problem

Did you lot know?

Vectors are very useful and tin be extended beyond GCSE mathematics. Vector analysis is the branch of mathematics that studies vectors.

At GCSE we study two-dimensional vectors, simply we can also look at three-dimensional vectors.

In A Level maths cartesian coordinates are likewise referred to as position vectors when we employ a coordinate system as our vector space. In maths a vector is an element of a vector space.

Vectors tin as well be extended farther past learning how to multiply two vectors together using the dot product.  This is too known equally the scalar product of two vectors.  It is possible to multiply vectors and this is known as a cross product.  This is as well known as the vector product of two vectors.

Yet stuck?

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How To Write Vector Notation,

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