Lesson video

Hi there, my name's Ms. Lambell.

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

Let's get going.

Welcome to today's lesson.

The title of today's lesson is Fluency in Arithmetic with Vectors, and that's within the unit, Vectors.

What will we be able to do? Well by the end of this lesson, you will be able to carry out arithmetic procedures on vectors and also draw a resultant vector.

Keywords that we'll be using in today's lesson are vector, displacement and resultant vector, all of which should be familiar to you, but it's always worth, I think, having a quick recap, just to make sure we are happy before we move on.

A vector can be used to describe a translation.

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

Remember the top number tells you your horizontal displacement and the bottom number tells you the vertical displacement.

The top number, if it's negative, we move left, positive, we move right, and the bottom number, if it's negative, we move down, and 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 and you will have been familiar with that, from other work that you've been doing.

Here we go then.

Today's lesson in two separate learning cycles.

First one, we will concentrate on just working with operations and vectors, and in the second one we'll extend that to looking at vectors on a grid.

Let's get going with the first one.

So combining operations with vectors.

We need to find the resultant vector of three, two, negative three, add two, three, negative one.

First thing we're going to do, is we're going to work out what the resultant vector of this first bit is.

Here we've got a scalar and we've got a vector.

What we need to do to find the result of vector is to find the product of the scalar and the two original components.

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

We are going to do exactly the same now with the second vector, two, three, negative one.

We're going to find the product of the scalar and the original components.

Two multiplied by three is six and two multiplied by negative one is negative two.

We were finding the sum of those two things, so we're going to take each of those separate resultant vectors, and then we're going to find the resultant vector of those two.

Six add six is 12, and negative nine add negative two is negative 11.

So I find it easier if I separate my two original vectors and then add them back together at the end.

Let's do that again then, but with a different question.

So I'm going to take my four with vector negative three, two.

So remember that's saying four lots of that vector, so the magnitude is gonna be four times the size.

Four multiplied by negative three is negative 12 and four multiplied by two is eight.

We do the same now with five and then the column vector six, negative one.

Five multiplied by six equals 30 and five multiplied by negative one is negative five.

We look back to the question, the question was asking us to find the sum of those two.

So we take our separate column vectors and we add them, negative 12 add 30 is 18, and eight add negative five is three, the resultant vector is 18, three.

Let's take a look at what happens if we're finding the difference.

Again, we're going to multiply the scalar by the original components of the vector.

Two multiplied by one is two and two multiplied by four is eight.

And then I'm gonna deal with the second part.

Negative three is my scalar, so negative three multiplied by negative two is six, and negative three multiplied by five is negative 15.

We're going to add those two together, and the reason we're gonna add those two together, is 'cause we've multiplied by the negative three.

We've dealt with that already.

So two add six is eight, and eight add negative 15 is negative seven.

The resultant vector is eight negative seven.

Now your turn.

I'd like you please to find the resultant vector for that calculation.

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

Great work and you should have resultant vector of five, negative two, three is negative 10, 15.

The second one, don't forget that that's a negative two.

So negative two multiplied by negative three is six and negative two multiplied by negative four is eight.

And then we find the sum of those two.

Negative 10 add six is negative four, and 15 add eight is 23.

The resultant vector is negative 4, 23.

Here we've got the resultant vector.

So we're given the resultant vector is negative two, 14, and we are given some information about some vectors and some scalars, but we are missing a scalar here.

I'm going to use the horizontal displacement to form an equation and solve to find the value of A.

So I'm just looking at the top line, the horizontal displacement, and I end up with 3A.

A multiplied by three is 3A, add four multiplied by negative two.

And we know that that is equivalent to negative two, 'cause that is the horizontal displacement in the resultant vector.

And then if we simplify this, we end up with three A subtract eight equals negative two, because four multiplied by negative two is negative eight.

Then we're going to add eight to both sides of the equation and we get three A equals six, and then we're gonna divide both sides of the equation by three to give us A equals two.

Now here I want us to always check.

We can be confident that our answer is right if we use this check.

We know that the vertical displacement has to use the same value of A, so I'm going to now assume that I'm correct and that A is two, and I'm going to check that I end up with a resultant vertical movement of 14.

Two, which is what we've worked out A is, multiplied by negative five, so I'm looking at that first column vector with its scalar and I'm adding four multiplied by six.

If I do that, two multiplied by negative five is negative 10, add four multiplied by six is 24, negative 10 add 24 is indeed 14.

So I can be confident that I have calculated the correct value for A.

Let's take a look at another one.

Again, I'm going to use the horizontal displacement to form an equation and solve to find the value of A.

Sometimes you might want to mix it up a bit, you might decide that you want to use the vertical, and then check using the horizontal.

It really doesn't matter.

Let's form our equation.

Three multiplied by two is six, negative A multiplied by six is negative 6A, and we know the result of that is going to be 24.

That's the overall horizontal displacement represented by the 24 in the resultant vector.

We're going to solve this equation.

Six subtract 24 equals 6A.

So I've subtracted 24 from both sides and added 6A to both sides, and then I'm going to do six subtract 24 is negative 18, equals 6A, then I'm gonna divide both sides of my equation by six, giving me A equals negative three.

And I want to see you doing this check please.

We're gonna check using the vertical displacement.

Three multiplied by negative four, subtract what we think A is, which is negative three, multiplied by the vertical component in that second column vector of negative one.

Three multiplied by negative four is negative 12, negative three multiplied.

Because remember here, we need to think about order of operations.

We're gonna do our multiplication first.

Negative three multiplied by negative one is three, and it was finding the difference between those two.

So I end up with negative 12, subtract three, giving me negative 15, and I can then see that I must be right.

I've chosen the correct value of A.

Now it's your turn.

I'd like you to have a go at this one please, independently.

Take your time over it.

It's just about bringing together some bits of maths that you might not have used before in the same lesson, but you've got all of the skills you need to be successful at this.

So pause the video and then come back when you are done.

Super.

Let's check.

So your equation should be 6X subtract 10 equals 14, add 10 to both sides, we end up with 6X equals 24.

Divide both sides of the equation by six, we end up with X equals four.

Now please tell me that you did your check.

Yes, of course you did.

You made me happy.

We're gonna check using the vertical displacement.

We know that X is four or at least we think X is four.

So we end with four multiplied by one, subtract five multiplied by negative four, doing the multiplication first because of order of operations.

Four multiplied by one is four, negative five multiplied by negative four is 20, and that's positive 20, so now I can find the sum of those two, four add 20 is 24, and I can clearly see that I must have the correct answer.

Otherwise I'm not going to get that resultant value of 24.

Here you are going to be working out for me the punchline to the joke.

What is king of the pencil case? You will need to rearrange your letters once you've worked out all of the eight answers into a two word answer.

So you're going to work out the resultant vectors, find the answer in the grid, that will give you a letter.

You then need to rearrange the letters into two separate words to answer the punchline to the joke.

Pause the video and good luck.

Great work.

Question number two, given that the resultant vector is four, 19, I want you to find the value of A and the same in B, you're gonna find the value of A.

Remember to check your answer, using opposite displacement to the one you used to find the value of A in each.

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

Superb work, well done.

What is king of the pencil case? And it was the ruler.

I'm sure you are all absolutely rolling around on the floor laughing now, it's so funny.

So A, the answer was L, B was E, C, The answer was H, D was E, E, the answer was R, F was T, G was U and H was R.

Like I said, if you rearrange those letters, you end up with the ruler.

Question two, you should have ended up with an equation 2A subtracted two equals four, lead into 2A equals six, A equals three, and then B, you should have ended up with negative 12 add 7A equals negative 19, then 7A equals negative seven, A equals negative one.

Let's move on then to the second learning cycle, and we're going to look at what happens when we look at some of these things on a grid.

Draw the vector two, two, negative three and three, negative two, four.

Let's multiply the scalar by each of the separate components.

That gives us four, six.

So let's draw that vector, then we'll do with the second part of the calculation.

So three, negative two, four, which is negative six, 12.

So I'm gonna go negative six, so six to the left, remember from the end point, and then I'm gonna go up 12.

Resultant vector remember, is the vector that takes us from the start point, which is my black dot, to our end point, which is the end of that arrow, which is where the green dot is, and then don't forget to join those points together with a ruled line, with an arrow head on, to show the direction.

That is the vector.

Okay, and another one.

Deal with each part separately, three multiplied by two equals six, three multiplied by negative one is negative three.

We're gonna draw that.

Six negative three, six right, and three down.

Then we're gonna deal with the second part, which is negative two multiplied by four, which is negative eight, and negative two multiplied by five, which is negative 10.

So for my final point, I now need to go negative eight, so eight left and 10 down.

And then my resultant vector is joining the black dot to the green dot, and I can see now that that is my resultant vector.

Remember the resultant vector takes you from the most direct route from the start to the finish.

Let's do one more together.

This is gonna be five multiplied by two is 10, and five multiplied by negative one is negative five.

Let's draw that on.

So it's gonna be 10 right and five down.

What is the scalar here? The scalar is negative one, so you might want to put that one in.

Negative one multiplied by negative six is six, and negative one multiplied by 10 is negative 10.

Let's draw that on.

Six to the right and 10 down.

Now let's draw on the resultant vector, so from the black dot to the green dot, and then I can clearly see that that is my resultant vector.

Here is Lucas's attempt at drawing the vector, three, negative two, four, subtract two, four, negative one.

Got some working out, and then we've got some drawing.

I'm going to give you a moment.

I'm gonna ask you to pause the video and to look carefully through what Lucas has done, and when you can see what he's done, come back, and we'll see what Sofia and Lucas are talking about.

Sofia says, "Hmm, I'm not sure.

I think you've made two errors in your drawing." Lucas says, "Oh no! Where do you think the errors are?" Sofia says, "Your vector is going in the wrong direction." Lucas says, "Oh yes! I have drawn it from the end point to the start, and it should be the other way round." It's really important that your vector, your arrow, points in the direction taking you from your start point to your end point.

And Lucas says, "Great, thank you.

But what about the other mistake?" Sofia says, "I always do a double check." So we've got our two separate vectors, negative six, 12, and negative eight, two, and she's added those together, and that gives negative 14, 14.

Now Lucas says, "If I draw the correct arrow pointing in the other direction, my vector is negative 14, 10." So he's noticed that he's gone 14 left, but only 10 up when it should be 14 up.

So actually we should have gone 14 left and 14 up.

That's what the vector should have been.

So again, that idea of checking your answer, if there's any way you can check your answer, make sure that you use it.

Lucas says, "Thank you for your help, Sofia.

When drawing the vector, I went in the wrong direction." And Sofia says, "You are welcome Lucas." "So, surely I could have found the resultant vector and just drawn that?" That was what Sofia did to check Lucas's answer, didn't she? And Sofia says, "Yes, in that question you could.

But if the question had said, 'show that,' you need to show the vectors you are finding the sum of." So really important, show that question, is you must actually show your original vectors and then how that gives you the resultant vector.

Your turn to have a go at this question.

A to B is represented by the vector two, five, three, and B to C is represented by the vector three, negative two.

Sorry, negative two, one.

Show that A to C is represented by the vector four, nine.

Pause the video, give it a go, and then call back when you are done.

Let's take a look then.

The first one, two multiplied by five is 10, 10 to the right, and two multiplied by three is six, so six up.

That now gives us point B.

B to C, so to go from B to C, we're gonna use the vector three, negative one, two.

Three multiplied by negative two is negative six, that is six places left, and three multiplied by one, it's three, so that's three up.

That gives me where point C is.

Now we can join A to C and check that is represented by the vector four, nine.

So A to C is four across to the right and nine up.

So yes, we have shown that A C is represented by the vector four, nine.

And finally task B, you're gonna draw the vectors, and I've given you the grids, and I've given you some starting points.

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

Okay, and question three, a show that question.

So you can't just work out the resultant vector and draw it on, you need to show me please what the resultant vector of that calculation is.

And here are your answers.

Here I would suggest that you pause the video and check your answers.

The grey line's the original two vectors, and the black line is the resultant vector.

And question number three, again, the grey lines are the original vectors and the black line is the resultant vector.

Summarising our learning from today then.

Vectors can be multiplied by a scalar.

So for example, five, negative one, two, five multiplied by negative one is negative five, and five multiplied by two is 10.

Arithmetic procedures can be applied to vectors.

Remember, you must make sure that you consider order of operations, and there is an example there of one that we looked at during this lesson.

Resultant vectors can be represented on a grid.

You can find the resultant vector and draw this, or find the sum of the vectors and represent this on the grid.

If however, a question says, "show that," you must draw the original vectors and not just the resultant vector.

Well done on today's learning.

You've done fantastically well and I've enjoyed working alongside you.

Hopefully I will see you again really soon to do some more maths.

Take care of yourself, goodbye.