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 "Subtraction with Vectors" and that's within the unit Vectors.

By the end of this lesson, you will be able to find the differences between vectors.

Some keywords we'll be using in today's lesson are vector, displacement, resultant vector, and additive inverse.

A vector can be used to describe a translation.

The vector two negative five shows the translation of two units right and five units down.

Remember the top number tells us the horizontal displacement and the bottom number tells us the vertical displacement.

If the top number is negative, we move left, if it's positive, right, and if the bottom number is negative, we move down, or if it's positive we move up.

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

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

The additive inverse, this is something that you might not have looked at for a little while.

So the additive inverse of a number is a number that when added to the original number, gives a sum of zero.

Today's lesson is split into two separate learning cycles.

In the first one, we will concentrate on subtracting column vectors.

In the second one, we will then move on to looking at what happens if we represent these on a grid.

Let's get going with the first one, so subtracting column vectors.

What is the additive inverse of the vector three eight? And Sam reminds us the additive inverse makes the sum of two numbers zero.

What will the resultant vector be for an additive inverse? Sam says, "It must be zero, zero." Is that what you said? Of course you did 'cause Sam's right and so are you.

What displacement does the vector zero, zero represent? It represents no displacement at all.

It shows zero displacement horizontally and zero displacement vertically.

Sam says, "So if this vector means three right and eight up, we must have to reverse that to end up back where we started." What do you think the additive inverse of vector three, eight is? the additive inverse of the vector three, eight is negative three, negative eight.

What do you think the additive inverses are of the following? I'm going to give you a moment to take a look at these and decide if you can find what the additive inverses are.

You may want to pause the video if you need a little bit more time.

Remember, the additive inverse means we want the sum of the two to be zero, therefore we need the resultant vector to be zero, zero.

What do I do to one to make it zero? I subtract one.

So the additive inverse of one is negative one and the additive inverse of four is negative four.

So we can just write this in our column form.

Second one, what did you come up with? Of course you came up with negative four, two.

And the third one? Yep, that's right, it's two, three.

And the final one would be, you got it five, negative seven.

I think you'll be able to spot a really quick way of working out what the additive inverses are.

Well let's take a look at the graphical representation of these two vectors, four, negative two.

There is my vector four, negative two.

From a starting point, I've moved four places to the right and two places down.

Now let's take a look at the vector negative four, two.

Negative four, two.

I've moved four places to the left and two places up.

Now we know that the overall resultant vector is zero, zero.

What is the same and what is different about the two vectors? They are parallel, we can see that they're parallel and they have the same magnitude because they are the same length, but they are in opposite directions.

Notice the arrows are pointing in opposite directions.

The additive inverse of a vector will be the same vector in terms of its length and it will be parallel.

It will just be moving in the opposite direction.

We know that additive inverses allow us to write any subtraction as an addition of additive inverses.

Let's just start off with some straightforward calculations involving just one number.

Four, subtract nine.

The additive inverse of positive nine, 'cause here we could read this as four subtract positive nine is negative nine.

So we would rewrite this as four add negative nine.

I'm rewriting my subtraction as an addition.

Negative four, subtract nine.

That nine is positive.

So therefore we are going to add the additive inverse of positive nine, which is negative nine.

We end up with negative four, add negative nine.

Negative nine, subtract four.

Here, this is going to be negative nine.

And we're going to add the additive inverse of four, which is negative four.

Negative four, subtract negative nine.

Here we are going to add to negative four, the additive inverse of negative nine, which is positive nine.

We can do exactly the same with our column vectors.

It's just treating the top and the bottom of the column vector separately.

I'd like you please to match each of the subtractions to its correct equivalent addition on the right hand side, pause the video and then come back when you are done.

How did you get on? So, first one matches with the third one.

The second one matches with the first one.

And then the third one matches with the bottom one.

And then finally the bottom one matches with the second option.

How did you get on? Great work, well done.

Let's take a look at this calculation.

We're finding the difference between the vector two, three and the vector six, eight.

First thing we're going to do is we're going to write the subtraction as an addition of the additive inverse.

Two, three, my initial column vector doesn't change, but I'm gonna add the additive inverse of six, eight, which is negative six, negative eight.

Now we we used to add in together vectors.

We're just going to find the total displacement horizontally and vertically.

Two, add negative six is negative four and three add negative eight is negative five.

This remember, is the resultant vector.

If I move along the vectors two, three and negative six, negative eight, if I actually go in one straight line to get from the start to the finish, it would be negative four, negative five, and that's the resultant vector.

Let's take a look at this one, negative, one six, subtract negative three, negative two.

We're going to rewrite the subtraction as an addition of the additive inverse.

Why don't you give this one a go before I put the answer up? And you should have negative one, six, add the additive inverse of negative three, negative two, is three, two.

Negative one, add three is two and six add two is eight.

By writing it as a addition of the additive inverse, allows us to do those calculations quickly.

And now a matching activity, you're gonna match each of the questions on the left hand side with the resultant vector.

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

Option one matches with option four.

So the first one is negative eight, negative four.

The second one is four, negative eight.

The third one is negative eight, four.

The fourth one is negative four, negative eight.

The fifth one is eight, four and the final one is negative four, eight.

Now you may need to pause the video 'cause I went through those quite quickly just to check.

And also it's a little bit confusing with all of the lines, so just make sure you follow those lines carefully to make sure that you've made any errors, but of course I know you haven't made any errors.

You are ready now then to have a go at task A.

In question number one, I'd like you please to find the resultant vectors for the following.

So you're going to pause the video.

Remember, I'd like you to rewrite any subtraction as an addition of the additive inverse.

And then when you've done that, you can come back, and we'll move on to question number two.

Good luck with these, you can pause the video now.

Great work, now let's move on to question number two.

Find the value of A in the following.

So you're gonna find the value of A, I've given you the resultant vector, but I've left out a component of one of the original vectors.

Really important here, I would always, once I've got my answer, it's just double check that it does give that resultant vector.

So pause video, good luck, and I'll be waiting for you when you get back.

Great work, let's check those answers for you.

Question number one, A is the vector two, negative six, B, the vector eight, negative 16, C, the vector eight, sorry, one, negative eight, D, the vector eight, negative one, and E was the vector zero, negative one.

Well done if you've got all of those right.

Question two, A was four, B, negative seven, C, negative five, D, two, and E is negative two.

How did you get on? Of course you've got 10 out of 10.

Brilliant, we're ready to move on then.

Like I said, in this second learning cycle, we're going to concentrate on subtracting vectors, but we're gonna represent these on a grid.

We need to draw a representation to show that the following is true.

The vector three, four, subtract the vector three, one, the resultant vector of that is zero, three.

We're gonna rewrite as an addition.

So we're going to add the additive inverse of the vector three, one, which is negative three, negative one.

Now let's draw this.

Three, four, and I'm gonna add on negative three, negative one.

So let's check.

My first vector told me to move three places to the right and four places up.

And then from that point, my second vector says I need to move three places to the left and one place down.

Let's look for that resultant vector, which is going from the start point to the end point in one straight line.

And then we can see that we've moved zero horizontally and we've moved three up.

So we can see we've shown that that is true.

Let's try this one.

Starting with the vector four, three, four to the right and three up.

But before we do that, I nearly forgot, we're gonna rewrite our subtraction as an addition.

Four, three, add the additive inverse of negative two, four, which is two, negative four.

Now I'm going to draw my vector of four, three, four right, three up.

I'm gonna add to that the vector two, negative four, so two to the right and four down.

And then I'm going to join together my starting point with my endpoint and I'm going to have a look and I'm gonna check that my vector that takes me directly from my start to my endpoint is six, negative one, six to the right and one down.

It is, we've shown it, we've shown that that is true.

And another one, this time I won't forget, rewrite as an addition.

Negative three, add two, four.

And we need to show that that equals negative one, six.

Let's start with our first vector, negative three, two, three left, two up.

Now we're gonna add on to that the vector two, four, two to the right and four up.

And then we're going to find the resultant vector, which is going from the green dot from our start point to our endpoint in one go.

I've gone one left represented by negative one and I've gone six up represented by the six.

So yes, we've shown that that is true and let's have a look at this one.

So exactly the same process.

I'm going to make sure I rewrite it as an addition.

I'm going to draw my vectors and then I'm going to draw the resultant vector, which is my black line.

And then I'm going to check that is three to the right and it should be eight down and it is.

We'll do one more together.

And then you are more than capable then of having a go at one independently.

Draw a representation to show that the following is true.

And I've given a starting point here.

Firstly, we're gonna rewrite the subtraction as an addition.

So we end up with the same starting column vector.

I put where I add in the additive inverse of negative four, negative three, which is four, three.

From the dot, I need to move two places left and zero vertically.

So that takes me to that point.

And then I'm going to move four right, and three up, which takes me here.

And then I need to draw on my resultant vector, which is from the start, which is the green dot to the end.

And then I just need to check.

The resultant vector I was told was two, three.

The resultant vector on my diagram I can see is two to the right and three up.

Over to you, pause the video, give this one a go, and then come back when you're done.

Great work, hopefully you remembered to write the subtraction as an addition of the additive inverse.

So we're going to add six, negative four instead of subtracting negative six, four.

We're gonna draw that first vector so it is zero horizontally and three down.

From that point we're going to draw the vector six, four, which is six, right, and four down.

Then my resultant vector from the start all the way to the end point.

And then we just need to check, have I moved six places right and seven places down.

And I have, so we have shown that the following was true.

Task B now then.

You are going to draw representations to show that the following are true.

So just like we did in those examples and the one you've just done independently, I'd like you please to pause the video and show that the following are true using the grids please.

Great, well done, and C and D, same thing.

Show that the following are true.

All done? Here we go then with our answers.

We can go our starting point, we are gonna rewrite it as an addition.

So we would end up, if I had written that down, sorry, is negative five, three, add negative two, negative four.

So I started at my dot, I moved five places left and three places up.

And then from there, I went two places left and four places down, giving me the overall resultant vector of negative seven, negative one.

The second one, you should have negative two, negative five, add four, negative two.

From the starting point then, we're going to move two places left, five places down.

And then from there, we are going to move four places right, two places down.

And we can see that that is equivalent to the vector two, negative seven.

From start to finish, I have moved two right and seven down.

Part C, gonna have negative five, negative two, add one, five.

So this is my starting point.

I'm going to move five places to the left, two places down.

And then from that endpoint, I'm going to move one place right and five up, draw in my resultant vector, which is the solid black line, and then check that that is a movement of four places left and three places up.

And yes it is.

And finally the last one, I'm going to do minus six, three, add four, negative seven.

I'm then going to draw those vectors.

I'm starting with the vector negative six, three.

So I'll move six places left, three places up.

And then I'm going to add on the vector four, negative seven.

So from the end of that point, I'm going to move four right and seven down.

Drawing on the solid line, which is my vector, my resultant vector.

And I can see that that indeed is a movement of two to the left and four down.

Great work, now let's summarise our learning from today's lesson.

When finding the difference between column vectors.

Firstly, we're going to rewrite the subtraction as an equivalent addition.

And we do this by using additive inverses and we can then find the sum of the horizontal displacement and the vertical displacement to give the total overall displacement.

So for example, the column vector negative three, two, subtract the column vector negative two, negative four.

We're going to rewrite that subtraction as an addition.

The additive inverse of negative two, negative four is two, four.

We then are going to find the sum of the horizontal displacement and the sum of the vertical displacement.

Negative three add two is negative one and two add four is six.

And remember, that gives us the resultant vector.

Superb work today, well done.

I've really enjoyed working alongside you with finding differences between vectors and I hope to see you again really soon.

Take care of yourself, goodbye.