Lesson video

Hi there, my name is Miss Lambell.

You've made a superb choice to deciding to join me today to do some maths.

Let's get going.

Welcome to today's lesson.

The title of today's lesson is Algebraic Vector Notation, and that's within the unit vectors.

By the end of this lesson, you'll be able to represent information graphically given algebraic vector notation.

Keywords that we'll be using in today's lesson are vector, displacement and resultant vector.

You should be really familiar with all of these now.

A vector can be used to describe a translation.

And the vector 2, negative 5 shows a translation two units right and five units down.

Remember, the top number tells you the horizontal displacement.

If the value is positive, the displacement is moving to the right.

And if it's negative, it's moving to the left.

The bottom number in the column vector tells us the vertical movement.

If it is a positive number, it is going to move up and a negative number is going to move down.

The way I remember this is that that's where the negative numbers are on my set of axes.

If I draw them, they are to the left and they are down the bottom.

Displacement is the distance from a starting point when measured in a straight line.

That's really important when we're doing lots and lots of vector work.

A resultant vector is the single vector that produces the same effect as a combination of other vectors.

Effectively, it's the shortest distance between two points, and we can work this out using known vectors.

This lesson is split into two learning cycles.

In the first one, I will introduce you to vector notation, different ways that vectors can be represented.

And in the second one, we will concentrate on what we're calling vector pathways.

You should find today's lesson fairly straightforward.

Let's get going with that first one then.

So, let's have a look at what we mean by vector notation.

Here we go.

Notation in mathematics is very important, extremely important.

We use symbol and notation because they are easier to read and to understand.

They are concise and take up less space.

They can be used to represent complex concepts.

It allows mathematical ideas to be communicated more effectively than words.

If we think about it, when we are writing three multiplied by four, we use that multiplication symbol to save us writing the word multiplication.

When we're dividing, we use the division symbol.

When we are repeatedly multiplying, we use an index.

Maths is much easier to communicate if we use notation.

This notation, however, needs to be standardised so that everybody knows what we're talking about.

Here we have a vector from A to B.

How do I know the vector is travelling from A to B? That's right, because that's the way that the arrow is pointing.

We're travelling from A to B.

We can write this vector in column form, and you'll be very familiar with doing that.

Column form here, we can see that moving from A to B, my horizontal displacement is one to the right, represented by the one at the top of my column vector, and the three represents my movement of three up in the vertical direction.

We can also write this vector using the points A and B and indicating the direction with an arrow across the top of the letters.

The vector A, B is equal to 1, 3.

We can also use algebraic notation.

So, we see here the letter A.

Notice it's lowercase.

My points at the end of my lines are labelled with uppercase letters and my vectors will be labelled with lowercase letters.

You can see here that actually it's a bold A.

So, we could write that A equals 1, 3.

We can use algebraic notation.

There are variations using algebraic form.

You may have A with a line underneath or a squiggle.

Sometimes you may have A with an arrow across the top, or you may have a bold A.

The bolded letter is the most common way of representing a vector.

So, if you see a bold lowercase letter, it's going to be representing a vector.

Let's look at linking all of these forms together.

Here we have O, B.

We can see that we're going from O to B and that is the vector 1, 3.

It's also labelled with a lowercase letter B, which we can see is bold.

We could also therefore say that B equals 1, 3.

What would the notation look like if I wanted to multiply this vector by 3? It would look like this.

It's parallel.

It's travelling in the same direction.

It's three times the length.

This I would then write as 3, O, B.

So, I started with O, B.

Effectively, it's three lots of O, B like we're multiplying.

And we can see that the resultant vector of that is 3 to the right and 9 up, represented by the column vector 3, 9.

Or we could write it as 3, B.

I've used three of the B vectors to make the total length of the vector 3, B.

So, these are all ways of showing exactly the same thing.

I'd like you to have a go at this check for understanding, please.

I've given you here a table.

You've got your algebraic form, and you've got your column form, and then vector notation.

Each of the vectors is starting from O, the centre.

I'd like you, please, to fill in the missing information.

So, pause the video and then when you're ready, come back and we'll check those answers for you.

Great work.

Let's have a look.

We can see that the bold letter A is next to the vector notation OA.

It's the line joining O to A.

And to get from O to A, I've moved 2 right and 2 up, giving me the column form of 2, 2.

Which vector has moved from the centre O, 3 left and 2 up? Because that's what the vector negative 3, 2 is telling us.

And that is the vector Q.

Q is the vector joining O and B.

So that's the missing vector notation here.

Right, OC.

Let's have a look.

What is the algebraic form for the vector joining O to C? And that's R.

What's the column vector to get from O to C? I'm going 3 left and 5 down.

So that's negative 3, negative 5.

And then there's only one left.

We've done A, we've done Q, we've done R.

So, we haven't done P.

P is joining O to D and is 3 to the right and 2 down.

And that completes the table.

How did you get on? Well done.

Here we have the vector AB.

It can be written as AB equals B, because we can see the algebraic notation of vector AB is B.

AB is equal to the column vector 3, 6.

We can see that represented on the grid.

Given that, how would you write the vector BA? BA is going in the opposite direction.

We're going in the opposite direction, and therefore it is going to be the negative of vector B.

BA is equal to negative B.

And BA is equal to negative 3, negative 6.

That's the column vector representing BA.

So, if we move in the opposite direction, then the vector becomes a negative.

Write the algebraic forms of the other two vectors in terms of vector A.

Pause the video, you've got two to do, and I'll be waiting when you get back.

Okay, and how did you get on? You should have, for the first one, it's parallel, so it must be linked to vector A.

It's travelling in the same direction, and it's twice the length.

So, this is the line 2A, or the vector 2A, I should say.

If we look at the second one, it is parallel to A.

It is the same length as A, but it is travelling in the other direction.

So, this is minus A.

And now this one.

Give this one a go.

Let's take a look at that, then.

So, B, we can see that the middle line is parallel to B.

It's travelling in the same direction, but it's three times the length, so it's 3B.

And if we look at the line on the right-hand side of the screen, we've got it's parallel to B.

It's twice the length, but it's in the opposite direction, so it is negative 2B.

Given we have vectors A and B, what do you think the resultant vector would be in algebraic form? And I've represented here the resultant vector with the purple vector line.

What do you think? The resultant vector would be A and B.

To get from the left-hand side of the purple line to the right-hand side of the purple line, I could do that by travelling along vector A and then vector B.

And there is no difference if we remove the grid.

We do not need the column form to identify the resultant vector.

Here we can clearly see that the result of going from one end of the purple line to the other is the same as travelling along vector A and then along vector B.

OC is the resultant vector.

Write OC in terms of A and B.

Pause the video, write down your answers, and then come back when you're ready.

Great work.

A, what did you get for A? To get from O to C, I travel along vector A and vector B.

So, it's A and B.

On B, how do I get from O to C? I travel along vector 2b, and then vector 3a.

Often you will see this written down.

If you check your answer maybe in the back of a book, you may see this answer written in alphabetical order.

So often people write them in alphabetical order.

So 3a and 2b.

Either of them are correct.

They're equivalents of each other.

And if we take a look at the final one, how do I get from O to C? Well, I travel from O to D, which is vector A, and then I'm going the wrong way.

I'm going against vector B, effectively.

I'm going the wrong way down a one-way street is how I think of it.

And so, it's A subtract B.

You're ready now to have a go at task A.

You need to pair the statements with the correct notation.

Pause the video and come back when you're done.

Great work.

And question number two.

You've got to fill in the blanks using the cards at the bottom of the screen.

Not all of them can be used, so some of them will be left over at the end.

Pause the video, decide what goes in each box, and then come back when you're done.

And question number three.

Here we have vectors P and Q.

I'd like you to write the resultant vectors in terms of P and Q, and the resultant vectors are shown with the purple dotted lines.

And finally, question number four.

OC is a resultant vector.

Write OC in terms of AB.

And here we can see that OC is the purple line.

Great work.

Let's check those answers, and then we'll move on to our final learning cycle for today's lesson.

AB.

This shows the line segment AB.

A, B in brackets is the coordinate pair.

AB with a line, sorry, an arrow across the top is the vector AB.

And AB in a column vector is, no surprises, a column vector.

Question two.

You've got to fill in the blanks.

A line segment starting from A and finishing at B can be written as AB.

A vector starting at A and finishing at B can be written in the form AB with the arrow across the top.

If given the algebraic form B, then 3AB equals 3b.

A vector starting at B and finishing at A can be written in the form BA with the arrow across the top, and is written algebraically as negative B.

Remember, because we're going in the opposite direction.

Question three.

The answers here are AQ add P, and B is Q add 2p.

And question number four, A, A add B.

B, 4a add 7b, and C, 3a subtract 2b.

Well done on those.

Now let's move on to the second learning cycle for today, and that is writing vector pathways.

Forming a vector pathway is very important because it structures your working out, it ensures that the working out is clear and easy to follow, and it uses correct mathematics notation for effective workings.

We need to write a vector pathway to represent the vector AC.

Unfortunately, there is no line segment joining A and C.

I like to think of it as a roadmap.

There is no direct route from town A to town C, so we can get from A to C, but we have to go via town B.

How else can you get from A to C using the vectors given in the diagram? The vectors, remember, are given as the lines with the arrows on.

To get from A to C, I need to go firstly from A to B, and then from B to C.

AC, the vector AC, is equivalent to the pathway vector AB and vector BC.

Let's take a look at another one.

Again, there is no line segment joining A and D, so I cannot go directly from A to D.

But how else can we get from A to D using vectors given in the diagram? This time, I'm going to give you a chance to decide what your answer is before I go through it.

AD, to get from A to D, I need to firstly travel along AB, and then from there I need to travel from B to C, and there I need to travel from C to D.

The vector AD is equivalent to the pathway vector AB and vector BC and vector CD.

It is always a good idea to check that adjacent letters are the same and that the start and end points are correct.

Now, when I say adjacent letters, they have got an addition symbol, or subtraction symbol between them.

I can see here that I've got B and B, so that means that my first vector stops at B and my second one starts at B, which is what I need.

And then if I look here, I can see I've got C and C.

And then just check.

We were working out the vector AD.

My vector does start with A and end with D, so it's always worth doing that little double check.

Your turn now.

How else can we get from D to C? I'd like you please to write down the vector pathway for DC.

Pause the video and come back when you're done.

Superb work.

Well done.

Now let's check that answer.

To get from D to C, I need to firstly go from D to A, then from A to B, and then from B to C.

The vector pathway representing the vector DC is the vector DA, add vector AB, add vector BC.

Write a vector pathway to represent AC.

AC equals AB, add BC.

And Jacob says, hang on a minute, the diagram shows the vector CB, not the vector BC.

What relationship is there between vector BC and vector CB? Now hopefully you already knew this from previous learning.

And it's that BC is equivalent to negative CB.

It's the same vector.

It's just the opposite direction, which is why it's negative.

It's parallel.

It's exactly the same length.

It's got the same magnitude.

The direction is the only thing that's different.

That's why it's negative.

We know then that BC, vector BC, is equivalent to negative vector CB.

So now I can finish this problem off.

So, AC is equal to vector AB, add negative CB.

So that's our vector AC, and we can write it in either way.

We can tidy up the plus and the minus if we want to.

Let's just take a look at another one to make sure that we've really cemented that in our brains.

AD.

I've written down here the vector pathway that takes me from A to D.

AB.

Oh dear, I can see that actually my vector is not AB.

The arrow is pointed from B to A.

AB is equivalent to the vector negative BA.

We need to just then rewrite the vector AB as negative BA.

The other two vectors can stay the same because the arrows are pointing in the correct direction.

Now your turn.

Pause the video.

Write me down, please, the vector pathway to represent DC.

Make sure that you carefully check which way those vectors are travelling.

Good luck, and I'll be here when you get back so that we can check that answer for you.

DC.

To get from D to C, I go from D to A, A to B, and then B to C.

BC, however, is going in the opposite direction.

We need to therefore rewrite BC in terms of CB because that's the vector we've been given, so it's negative CB.

Meaning my final answer is vector DA, add vector AB, subtract vector CB.

And you may have plus minus CB there.

That's absolutely fine.

Final task of today's lesson then.

Firstly, for this question, I'd like you, please, to write a vector path for the following.

So, you need to write a vector path for each of those vectors.

Pause the video and when you're done, I will reveal question number two.

Well done.

And question number two, as promised.

And these are our final four questions for today's lesson.

So, you're almost there.

Well done.

Pause the video and then come back when you're done.

Great work.

Let's check our answers.

Question number one.

A, vector AC is equal to vector AB, add vector BC.

B, vector BG is equal to vector BC, add vector CF, add vector FG.

C, vector AF is equal to vector AB, add vector BC, add vector CF.

And D, is vector AE is equal to vector AB, add vector BC, add vector CD, add vector DE.

Lots of vectors in there.

What I'd like you to do is make sure that you always check that you've got those adjacent letters the same, and you're starting and finishing at the right place.

And question number two.

Okay, I got very confused with all my vectors there.

So, what I'm gonna do, I'm not gonna read these ones out, especially as there are some negatives in there as well.

So, I'm gonna ask you to pause the video and then check your answers really carefully.

And when you're done, come back, and we will summarise what we've done during today's lesson.

How did you get on? Well done.

Let's summarise our learning from today's lesson then.

We know that we can write a vector in column form.

In algebraic form, and remember that may be a bolded letter.

It may be a letter with an arrow over the top, or it may be a letter with a line underneath.

Or we can use the points at either end of the vector and an arrow to represent which direction we are travelling in.

Addition of vectors allows us to construct new vectors and a diagram can help to determine which vectors should be used.

And often it's useful to sometimes highlight as well the pathway that you are using.

Remember to always double check.

Are you moving in the same direction as the vector? Because if you're not, you'll need to make that vector negative.

Well done today.

Thank you very much for your time.

And I look forward to seeing you again soon.

Goodbye and take care of yourself.