Lesson video

Hi there.

My name's Ms. Lambell.

You've made a superb choice 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 Multiplication with Vectors, and that's within the unit vectors.

By the end of this lesson, you will be able to multiply with vectors.

You will, I promise you.

A key word that we'll be using in today's lesson that you should be really familiar with is displacement.

Remember, this is the distance from the starting point when measured in a straight line, we have a start point and an endpoint.

Basically the straight line that joins those two together, possibly a new word today that we are going to be using is scaler.

A scaler is a quantity that has a magnitude but not a direction, and that will become apparent as we work through this lesson as to what that means.

For today's lesson, I've split it into two learning cycles.

In the first one, we will look at using a grid, what does it look like on a grid to help us to visualise what is happening? So multiplication or multiplying vectors represented on a grid.

Once we've done that, we will then move on to multiplying column vectors.

And actually, I think you'll find this lesson super easy, so let's get going with it.

The first thing then we're gonna look at representing multiplying of vectors on a grid.

We're gonna draw the vector two, three.

I've started in the bottom left hand corner and remember the vector two three means I'm gonna move two places to the right and three places up.

I represent that vector with a solid line and I have put an arrow on to show the direction to show that I started in the bottom left corner and I moved to the top right corner.

That is the vector, two, three.

What about if I was to draw the vector 2, 2, 3.

That just means two of the vectors, two, three.

So I would just draw that again.

Now the purple line shows the vector 2, 2, 3, and if I wanted to draw the vector, 3, 2, 3, I would need to repeat that for a third time.

Now the purple line is representing the vector 3, 2, 3.

This vector A two, five represents A to B.

We need to find the value of A.

Well, let's take a look.

We know that the vector is two, five, so and we're going from A to B, so I'm going to draw my vector two, five.

I've moved two right, and five up.

I need to keep doing that until I arrive at point B.

So I've gonna, I've got one of those vectors, two of those vectors, three of those vectors.

So it takes three of the vectors two, five to go from A to B.

This means that A is equal to three.

Let's take a look at this one.

Again, we're finding the value of A, we're going from A to B this time our vector is negative one, three.

So starting at A, I'm going to move one left and three up.

That's the vector negative one, three.

I need to decide how many of those it takes me to get from A to B.

So there's 1, 2, 3, 4, 5.

That means that A has a value of five.

I need five of the vectors, negative one, three to get me from A to B.

Your term I'd like you please to do exactly what we've just done.

This is the vector, A negative five, two, and it represents A to B.

What is the value of a? Pause the video and come back when you can tell me what that value of A is.

Good luck.

Super well done.

You will have started by drawing on the vector negative five two.

So from A, we're moving five left and two up.

There's my vector.

How many of those do I need to get from A to B? One, two, I need three of them.

So the value of A is three.

Alex is posing a question.

Does a scaler have to be a positive integer? What do you think? A vector has magnitude.

Remember that's the size and it also has a direction.

A vector can be any length at all, so it's magnitude can be anything.

And a negative value, remember means that it would be in a different direction.

So no, a scaler does not have to be a positive integer.

What happens when we a vector by a negative scaler? Well, let's take a look.

I'm going to use the vector one one to help us visualise this and see what's happening.

Here is the vector 1, 1.

One square to the right, one square up.

This represents the vector of 2, 1, 1.

I've just drawn my vector of one, one twice.

Remember, it's going in the same direction, so it should create a straight line.

This is the vector of 3, 1, 1.

We can see what's happening.

The line started off diagonally across one square, then two and then three.

How will the vector 0, 1, 1 be represented? Actually, that will just be represented by a dot because it means that we have not moved anywhere.

We've stayed in the same place.

What will happen to the vectors as we move into negative scalers? They'll go in the opposite direction.

So negative 1, 1, 1.

Notice the same as one one.

The arrowhead is turn turned around.

It's going in the opposite direction.

Negative 2, 1, 1.

If we compare that to 2, 1, 1, we can see that it's parallel.

It has the same magnitude, it's the same length line.

It's just the direction is different.

So when we multiply by a negative, it means the direction of the vector changes.

What is the same and what is different about these two vectors? They are parallel.

They have the same magnitude, they have the same length.

Remember magnitude shows this is represented by the length.

Their directions are opposite to each other.

The top one is the vector three negative two, and the bottom one is the vector negative three two.

We could then write negative three, two as negative one, three negative two.

It is parallel to the vector three negative two.

It has the same magnitude as the vector three negative two, however, it is in the opposite direction.

We're going to draw the following.

Here I've shown you what the vector of the purple heart looks like.

We are now going to draw what the vector of two purple hearts looks like.

Two purple hearts.

It's going to be parallel.

It's going to be in the same direction, but it's gonna be two times the magnitude and so therefore, the gradient is going to be the same, but it's just going to be twice as long.

It doesn't matter where I draw that on my grid, okay? As long as it's twice the length and it is parallel.

Basically this one has a gradient of a half.

What about if I were to draw negative three hearts? This will be parallel.

It will be in the opposite direction.

Why, how do we know it'll be in the opposite direction? Because it's a negative scaler and it's going to be three times the magnitude.

So the length of the line is going to be three times as long.

This is my vector representing negative three hearts.

So notice it is three times the length.

The arrow is going the opposite direction because we had a negative scaler.

What about 0.

5 hearts? Well, this one is gonna be parallel.

It'll be in the same direction because it is a positive scaler, but it's only going be half the magnitude.

So it's going to be half the length.

This is the vector representing 0.

5, heart.

Have a go at this one.

It's your check for understanding without counting, how can you tell that one of these vectors is not a scaled version of the other? Pause the video, write down your answer or say it aloud and then come back when you're done.

And what have you decided? They are not parallel.

They're not parallel.

The distance between them as we move sort of to the right and up is getting larger.

They are not parallel.

Which is the correct representation of negative 0.

25 hearts if heart is represented by the vector I've given you in the top right corner? Make your decision A, B, or C? And you should have come up with C.

Actually here it was obvious it was C and that reason for that is because it was the only one that had changed direction, so it changed direction.

The negative 0.

25 means it's gonna change direction.

The scaler is negative and it was going to be a quarter of the length, so it was C.

Your turn now.

I'd like you pleased to have a go at drawing the following.

So I've given you the original vector of heart, cloud, moon and lightning, and I'd like you please for A, to draw three hearts, B negative cloud C, negative 0.

5 moons, D, 1.

5 clouds and E zero point, sorry, negative 0.

6, lightning bolts, pause the video and pop back when you're done.

Great work.

And question number two, I need you please to work out what the scaler is.

So work out the scaler.

So X for A and what is Y for B, pause the video and then come back when you are done.

Let's check those answers.

What I'm going to do here, I'm just gonna ask you to pause the video and I'm gonna ask you to carefully check that your vectors are correct.

Remember to check as well the direction, the direction is really, really important.

In question two, X had a value of four and in part B, Y had a value of three.

Now we can move on to the second learning cycle.

So we're multiplying column vectors.

Find the resultant vector of three, two, negative three.

Okay, should we see what Alex has got to say about this, right? Two negative three, agreed.

And then the resultant vector of three, lots of that is here.

That's three, two, negative three.

And let's have a look at what that in vector is.

So it was six to the right and it was nine down, so six, nine.

Alex says, I have noticed that you just multiply each component by the scaler.

Three multiplied by two equals six and three multiplied by negative three is negative nine.

Yeah, that looks right, doesn't it? On the grid we're going to show that four negative one two is equal to negative four, eight vector negative four eight negative one, two.

There's my vector and I need four of those.

Remember they need to go in the same direction and be exactly the same length.

So I've just taken that vector and replicated it four times.

The green line is the resultant vector, it's the vector four negative two, it's the vector four, negative one.

two.

Now let's check that is equivalent to negative four, eight.

So I've gone negative four because I've gone four to the left and I've gone eight up, so plus eight.

So we have shown that that statement is true.

The scaler of four means the displacement are four times as large.

If we look at it, I moved one place, the original vector moved me one place to the left and the scaler of four I multiplied that by four, meaning I moved four places to the left.

My original vector took me two places up and I would need to multiply that by four to give eight places up.

Your turn now.

On the grid, I'd like you please to show that three five negative three is equal to 15 negative nine.

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

Superb work, and this is what your diagram should look like.

So that is our original vector, the vector three and sorry, five negative three, five to the right, three down.

We will need three of those.

I then I'm checking, I've moved 15 to the right and nine down, giving me the resultant vector of 15 negative nine.

I've shown that those are equivalent.

Find the resultant vector of 2, 3, 7.

To find the components of the resultant vector, we find the product, the scaler, and the original components.

Total horizontal displacement is two multiplied by three, which is six.

And the total vertical displacement is two multiplied by seven, which is 14.

My resultant vector is 6, 14.

Here we're going to multiply each individual component of the original vector by the scaler, which in this case is four.

The horizontal displacement four multiplied by two is eight, and the vertical displacement four multiplied by negative five is negative 20.

Giving me my resultant vector of eight negative 20.

And this one, same thing again, we're going to find the product of the scaler and the original components.

Horizontal displacement is negative two multiplied by negative six, which is 12, and the vertical displacement is negative two multiplied by 12, which is negative 24.

My resultant vector is 12 negative 24.

Now you are going to match each scaler and vector to the correct resultant vector.

Pause the video, work out the answers, and then come back when you're done.

Super work.

So you should have the first one matches with negative 12 negative 18.

The second one matches 12 negative 18.

The third one matches with 12, 18 and the final one matches with negative 12, 18.

And how did you get on? Well done.

Now you are ready for the final task of today's lesson, and hopefully, you'll find this fun and interesting.

You are gonna start at the dot and you're gonna draw the vectors in order.

Each endpoint is the start point of the next vector.

Really important.

The picture is wholly symmetrical, it's symmetrical.

You need to fill in the missing scalers to complete the picture.

So you'll notice there that we've got four boxes, dotted boxes with missing scalers.

Your job once you've drawn up to that point, is to work out what that missing scaler needs to be so that your diagram ends up as being a symmetrical picture.

Pause the video and then when you are done, come back.

Make sure you use a pencil and a ruler so that if you make any errors, which I know you won't, but you'll be able to rub them out.

And then also make sure you are using your ruler to make those vectors look lovely and clear.

Pause the video now and I will be here waiting when you get back.

Super work, let's check that answer.

You should have, hopefully I drew it myself.

So I'm not a great artist, but what should look familiar to you as a butterfly and that's what it looks like.

And then the missing scalers were the first one was negative three, the second one was negative or negative one.

The third missing one was two, and then the final one on that bottom row, the missing scalar was negative two.

Okay, then now it's time to summarise our learning from today's lesson.

Vectors can be multiplied by a scaler.

For example, the resultant vector of four, two negative five is eight, negative 20.

To find the components of the resultant vector, we find the product of the scaler and the original components.

Total horizontal displacement for the example above would be four multiplied by two to give us eight.

And the total vertical displacement would be four multiplied by negative five giving us negative 20.

The resultant vector is eight negative 20.

Multiplying by a negative scaler means the vector will be in the opposite direction.

So for example, we could write column vector negative three, two as negative one, vector three negative two, and it is parallel to the vector three negative two.

It has the same magnitude as three negative two, however it is in the opposite direction.

Fantastic work today.

Hopefully, you enjoyed working through vectors with me and multiplication of vectors and I'm hoping that you'll decide to join me again really soon to do some more maths.

I look forward to that.

Make sure you take really good care of yourself.

Thank you so much again for joining me, goodbye.