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 Addition with Vectors and that's within the unit Vectors.

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

The keyword that you may not have come across before.

So I'm just going to quickly tell you about it.

And that's resultant vector.

A resultant vector is the single vector that produces the same effect as a combination of other vectors and that will become apparent as we go through today's lesson.

Other keywords that we'll be using in today's lesson are vector and displacement.

A vector can be used to describe a translation.

The vector two negative five shows the translation two units to the 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, it means we move to the left, if it's positive to the right and if the bottom number is negative, we're moving down and if it's positive we're moving up.

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

Effectively, the shortest distance between two points.

Here are our two learning cycles for today's lesson and they are adding column vectors and in the second learning cycle we will look at adding vectors but we're going to represent them on a grid.

Let's get going with that first one.

So adding column vectors.

Here we have two column vectors and we are going to add them.

We're going to find the sum of them.

What movement do each of these vectors above represent? The first one represents two units right and three units up, and the second one represents a movement of six units to the right and eight units up.

If we were to sum the two vectors, what we need to do is we need to find the total horizontal displacement and the total vertical displacement.

Now I'm pretty confident that you'll be able to answer these questions.

What is the total horizontal and vertical displacement? What did you decide? If we look at the horizontal displacement first, we can see that two units to the right, six units to the right is a total of eight units to the right.

And if we look at the vertical displacement, 3 units up and another 8 units up is a total of 11 units up.

We can then write this as a column vector.

So 8, 11, and this is what we call the resultant vector.

It is the result of adding the two original column vectors.

Let's take a look at another one.

So again, I'd like you to write down or just say out loud, what movement do each of these vectors above represent? The first one represents five units to the right and six units up.

And the second one represents zero units horizontally and four units up.

If we to sum the two vectors, remember we're finding the total horizontal and vertical displacements.

What do you think the total horizontal and vertical displacements are? So starting with that horizontal displacement, we can see that five units to the right and then zero units horizontally, in addition to that, is five units to the right.

And the vertical displacement six units up and another four units up means in total we've moved eight units up.

We now write this back into its column vector form which is 5, 10, and remember this is the resultant vector.

It's the result of those two vectors.

Here you can have a go now at this check for understanding.

I'd like you please to fill in the blanks and find the resultant vector of the vectors 10/0 and 3/1, the sum of those two.

Pause the video and then when you've got your answer come back.

Well done.

Let's check those answers then.

Let's take a look at the first column vector.

This represents 10, on the top, represents 10 units to the right and the zero on the bottom represents zero units vertically.

Then looking at the vector 3/1, that's three units to the right and one unit up.

The total horizontal and vertical displacement are 13 units to the right and one unit up giving us a resultant vector of 13, 1.

Is that what you got? Of course you did.

Well done.

Now let's take a look at this one.

2/-1 add -5/3.

I'd like you please to write down or say out what each movement of the vectors are representing? The first column vector is representing two units to the right and one unit down, and the second column vector is representing five units left and three units up.

If we sum the two vectors, remember that is finding the total horizontal and vertical displacements.

What do you think the total horizontal and vertical displacement is in this situation? Two units right and five units left.

That may be useful here to draw like a little diagram, which I'm going to do.

So I've got my starting point and I move two units to the right.

From the end point, I mustn't go back to start point.

I want to work out what the overall is.

So from where I am now, which is at the right hand end of that arrow, I'm going to move five units to the left.

Now how far away am I from my starting point and my starting point is that purple cross and I can see that I'm three to the left of the purple cross.

So the overall horizontal displacement is three units left.

Now let's take a look at the vertical displacement.

Again, we'll start with a cross and I'm going to go one unit down.

From my finishing position, that's the end of the arrow remember, I'm now going to move three units up.

What was the total displacement from the purple cross to where we've ended up? And that is we can clearly see two up.

So the vertical displacement is two up.

The resultant vector is negative three two.

Just take a look at one more of those.

-2/2 add -1/-3.

What movement do each of the vectors above represent? Two units left and two units up and one unit left and three units down.

The sum of the two vectors we know is the total horizontal and vertical displacement.

What do you think it is this time? Remember you could draw that little diagram to help you if you need to.

Two units left, one unit left.

Well that's straightforward isn't it? Two units to the left and another unit to the left.

I'm actually three units to the left.

Two units up and one unit down.

So I'm gonna go up two units and down three units.

So the overall effect is I've actually moved down one unit, so one unit down giving me the resultant vector of negative three negative one.

Your turn now just as you did with the previous check for understanding, I'd like you please to fill in the gaps and then work out the resultant vector.

You can pause the video now.

Super.

Let's check those answers then.

We've got three units right and four units down.

That's the column vector three negative four and five units left and two units up.

That's the column vector negative five two.

The total horizontal and vertical displacement.

So the total is two left for the horizontal displacement and it is two down for the vertical displacement.

We then need to write that as a column vector.

It's -2/-2, so the resultant vector of 3/-4 add -5/2 is -2/-2.

Here we've got some sums of vectors and we've also got the resultant vector.

Let's see what Aisha's saying.

Aisha says, "I have noticed that we could have just added the horizontal displacement and the vertical displacement to give the total displacement." Do you agree with Aisha? Aisha is right.

Let's check a few.

Two add six is eight.

So that's all right.

And 3 add 8 is 11.

Yeah, that works.

Let's check a different one.

Two add negative five is negative three and negative one add three is two.

And just one final check to be sure.

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

Aisha was correct.

Using that method, I'd like you now please to match each of the following to their resultant vectors.

Pause the video and then when you are ready come back and we'll check your answers.

Super work.

Let's check those answers then.

<v ->2/3 add -6/-7 is -8/-4.

- 3/-6 add 7/-2 is 4/-8.

</v> <v ->10/6 add 2/-2 is -8/4.

- 3/-9 add -1/1 is -4/-8.

</v> 3/-2 add 5/6 is 8/4 and the final one obviously matches with the last one, which is 1/-3 add -5/11 and that is -4/8.

Now obviously at this point you could pause the video if you got a little bit lost as we went through those, there was a lot to list, wasn't there? Now you can do task A, I'd like you please to find the resultant vectors for the following.

Pause the video and when you are ready you can come back and we'll move on to question number two.

Well done.

And question number two, a little bit more challenging here.

I'd like you please to find the value of a, pause the video and come back when you are done with those.

Great work.

Let's check our answers then.

Question number one A, 6/12.

B, 4/2.

C, -5/-6.

D, 2/-13 and E, was -6/-3.

Moving on to question two.

A, was eight.

B was negative three.

C, negative four, D was three and E was negative two.

And how did you get on? Great work, well done.

Now we can move on then to looking at adding vectors, but we are representing them this time on a grid.

Let's get going.

We're going to draw a representation to show that the following is true.

The vector three four add the vector three one is equal to the vector six five, so the resultant vector would be six five.

Let's start with three four.

Going to choose a point and then I'm going to draw the vector three four.

When we draw a vector, we draw it with an arrowhead on it to show the direction.

Notice I've not drawn on the horizontal and the vertical counting parts because otherwise my diagram gets a bit confused.

So you really need to try if you can so do that counting sort of like in your head and just put the end point.

Right I'm now needing to add onto that.

So at the end of that line I'm needing to add the vector three one.

So I'm gonna add the vector three one, which means I'm gonna go from the end of my vector, three to the right and one up.

We want to know the resultant vector.

So if I'd started at the green dot and finish up at the end, this is the vector I'm trying to find.

Notice I've joined it together with a ruled line with an arrow head on.

Now I can check.

Six right and five up.

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

Let's take a look at another one.

7/3, the vector 7/3.

Let's start by representing that on our grid.

So we're going to go seven right, and three up remembering trying not to draw those lines on.

Then from that point I need to go -2/4, so two left and four up and I'm then finding a resultant vector and I'm checking to go horizontally from the green dot to the end of my vector line.

I've gone five to the right and seven up so we can see that it actually is true.

And another one, start with the initial vector.

Then we're going to draw the other vector.

Remember it needs to be the end of the line.

We don't go back to that starting point because we're adding it onto the previous vector, - 2/-4.

So I'm gonna go two left and four down.

The vector that goes straight from the green dot to the end of that vector is represented by the black line.

And then we can check five left.

Yep, and two up.

Yes, we now know and we've shown that the following is true using our graphical representation and another one, -2/-5.

We start by representing that, then adding -3/7 and then the resultant vector and then we can check, negative five and two up.

We'll do one more together.

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

Represent the vector -2/0.

So this time I you've got a starting point, so -2/0 and then I'm adding -4/-3 and then I'm finding the resultant vector.

So from the starting point to the final endpoint and we can see that this is a movement of six left and three down.

And now your turn, pause the video, have a go at this question and come back when you're done.

Okay, how'd you get on? Super, well done.

Represent the vector 0/-3.

So that's a horizontal displacement of zero so it doesn't move anywhere horizontally and then three down, then six left and four up.

The resultant vector is from the start point to the end point and we can clearly see that that is six left and one up, Are AC and ED parallel? Okay, so we know that parallel lines are multiples of each other.

We're going to need to work out the resultant vector for AC and the resultant vector for ED.

Let's take a look.

AC, how do I get from A to C? Now I can't go directly from A to C because I don't know what that vector is, but what I do know is I know that it is the resultant vector of AB and BC.

'Cause if I travel from A to B to C, that's the same as going the set directly from A to C, AB is 3/2 and we're adding BC, which is 3/-4.

Now I can find my resultant vector which is 6/-2, three, add three is six.

Two add negative four is negative two.

ED now ED, again, we don't know what that vector is, but we do know that I can travel from E to D via C basically.

So I can go from E to D, which is the vector five three.

And then from C to D, which is the vector -2/-4.

I'm now gonna find my resultant vector, five add negative two is three and three add negative four is negative one.

My two vectors then are three negative one and six negative two.

I'm looking now to see whether one is a multiple of the other.

What's the multiplicative relationship between three and six? That's multiplied by two.

Let's check is negative one multiplied by two negative two? Yes.

Therefore we can say that they are parallel.

There is a multiplicative relationship which is the same for the horizontal and vertical displacements.

Here we have a show that question.

Show that AC and ED are not parallel.

We know they're not parallel.

So when we get to the end, if we end up with one being a multiple of the other, we know we've made an error.

That's why I like show that questions 'cause I know what I'm working towards and I know if I've made an error when I get to the end.

AC, now we look to go from A to C.

We can't go directly at the moment because we don't know what that vector is, but I can go from A to B and then from B to C.

AC then is the resultant vector of AB and BC.

AB is one four and BC is three two.

Let's find the resultant vector.

One add three is four and four add two is six.

ED.

ED, again, if we look, we don't know what that vector is, but we know that it's the same as the result of vector EC add CD.

EC is zero nine, CD is two negative three.

If I add those together, zero add two is two.

Nine add negative three is six.

Now we're looking to see if there is a multiplicative relationship which is constant for the horizontal and the vertical displacements.

Four, what do I do to get from four to two? I multiply by no 0.

5.

What do I do to get from six to six? I multiply by one.

Therefore they are not parallel as one is not a multiple of the other because the multiplier is different for the horizontal and vertical displacements.

Are AC and ED parallel? AC is this vector and that's a resultant vector of AB and BC which is 0/2 add 2/2.

If I do that zero add two is two.

Two add two is four.

ED, ED is the resultant vector of EC and CD.

So I'm gonna add those two together.

Negative three add six is three.

And 10 add negative 4 is 6.

Then we're looking for that multiplicative relationship.

Two multiplied by something is three that's multiplied by 1.

5 and four multiplied by 1.

5 does indeed give a six.

Yes, AC and ED are parallel as there is a multiplicative relationship which is the same for both horizontal and vertical displacements.

Now I'd like you to do this check for understanding.

Which of the following vectors are parallel to 8 negative 12? So which ones are parallel? Pause the video.

Remember no guessing you should be writing down multiples.

You should be looking for those multipliers and then when you come back I will check those answers with you.

Well done.

Let's check those answers.

A, yes, it was.

If we look eight multiplied by 0.

5 is four and negative 12 multiplied by 0.

5 is negative 6.

B, yes, it was parallel.

This time the multiplicative relationship was negative two.

C, they were not parallel and D, they are and the multiplicative relationship was 1.

25.

Just remember that relationship, the multiplier does not have to be an integer and it can also be positive or negative.

Now you're ready to have a go at task B.

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

Pause the video and come back when you are done.

And question number two, bit harder here, justify whether AC and ED are parallel.

Pause the video if you need to go back and watch any of the examples remember you can.

That's absolutely fine.

And then when you've done and you've got an answer to this question or you've got to a point where you think, hmm, not sure what to do, come back and I'll go through the question with you.

And question number three is ABCE a trapezium? And again, that justification of your answer is essential.

You cannot just write yes or no and expect to get any credit for that.

You need to be able to back up and justify your answer.

Pause the video now and then come back and let me know if it's a trapezium or not.

Super work.

Well done.

Those last two were quite challenging, weren't they? How did you get on? Of course you did.

You did really well.

Here we go then.

One A, that is what your representation should look like.

Remember we were showing a representation on the grid to show that those statements were true.

And then B, the same thing.

You might need to just pause the video and carefully check your answers.

Question two, AC is a resultant vector of AB and BC, so we'd add those together, we end up with AC being seven five.

ED is a resultant vector of EC and CD, so if we find the resultant vector ED that is five three.

Then we are looking for that multiplicative relationship between the two horizontal movements, which is multiplied by five over seven and between the two vertical movements and the multiplier was 0.

6.

AC and ED are not parallel as one is not a multiple of the other.

Question three, we had to decide whether it was trapezium or not.

Now it is a trapezium.

AB is a resultant vector of AC and CB and that's means that AB is one six.

CE is the resultant vector of CD and DE.

So CE is negative 2, negative 12.

Then I'm looking to see whether those two lines are parallel.

Remember, a trapezium has one parallel sides.

So if this is parallel, then we can say is a trapezium.

One multiplied by negative two is negative two and six multiplied by negative 2 is negative 12.

AB and CE are parallel as there is a multiplicative relationship.

Therefore, ABCE is a trapezium.

Absolutely superb if you've got that last question right.

Now we can summarise our learning from today.

When adding column vectors, you find the sum of the horizontal displacement and vertical displacement to give the total displacement.

Remember, the resultant vector is effectively the sum of the column vectors.

Vectors can be added on a grid.

Remembering really important here, we've got that start point.

When you've drawn your first vector on, make sure that your next vector starts at the finishing point of the first vector.

And the resultant vector is the effect of going straight from the start point to the endpoint.

By finding resultant vectors, you can decide if vectors are parallel.

Parallel vectors have a multiplicative relationship.

And remember that multiplicative relationship needs to be the same for both the horizontal and vertical displacement.

Superb work.

As always, you've done fantastically well and I look forward to seeing you again really soon to do some more maths.

Like I said, fantastic work today.

Goodbye and make sure you take really good care of yourself.