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 "Column Vectors", and that's within the unit Vectors.

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

Keywords that will be used in today's lesson then are vector, displacement, and vertex.

A vector can be used to describe a translation.

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

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

A vertex is the point where two or more line segments meet, and the plural of this is vertices.

I'm pretty sure that you remembered all of those keywords, but it's always worth a quick recap.

Today's lesson, I've split into two separate learning cycles for you.

In the first one, we will look at column vectors from a graphical representation.

So I will give you the graphical representation and then you will write the column vectors.

And in the second part of the lesson, we will do it the other way around.

I will give you some column vectors, and I will be asking you to draw those with a graphical representation.

Let's get going with that first one, so column vectors from a graphical representation.

Describe the transformation.

We've got the object and we have the image.

Remember, the object is the starting position, and the image is the finishing position.

Notice that A becomes A prime, so that dash, A prime, B becomes B prime, and C becomes C prime.

Now you should be able to describe that transformation.

It's a translation.

We can see that A has moved horizontally right, and then up one to get to A prime.

The shortest distance to get from A to A prime is the black line, and the black arrow represents the vector, two, one.

What I could do is check this with the other two vertices, checking that B gets to B prime, two right, one up, which it does.

And C to C prime is two right, one up.

I know then that I have correctly translated that shape.

A vector can be used to describe a translation, a movement, and you'll be very familiar with that.

A vector has magnitude, size, so that's how long the line is, how big it is.

It also has a direction.

The length of the line shows its magnitude, and the arrowhead points in the direction.

Here we can see my purple arrow.

Its magnitude is the length of the line.

Its direction is which way the arrowhead is pointing.

So we can see that the arrowhead is pointing to the top right of the screen.

A vector is a straight line from the start to endpoint and covers the shortest distance.

The shortest way to get from A to B would be to draw a straight line connecting A and B.

The horizontal displacement is five right, and the vertical displacement is three up.

This is represented by the vector, the column vector, five, three.

The vector of A to B is five, three.

Let's take a look at this one.

We know the shortest distance between A and B is if I go directly from A to B.

This time I can see that my horizontal displacement is five left, and my vertical displacement is three up.

The vector of A to B is negative five, three.

Oh, let's take a look and see what Andeep and Sofia are chatting about.

Andeep says, "This is the vector, five, three." Sofia says, "This is the vector, negative five, negative three." Who do you agree with and why? Who did you decide was correct? And you should have said Sofia.

Sofia is correct.

Let's take a look at why Sofia is correct.

Now here I don't have my points labelled.

They're not labelled A and B.

Even if they were, I can't just assume that I start at A, I need to look at the direction that the vector is moving in, and I can see that it's going from top right to bottom left.

So I'm starting at the point at the top right.

My horizontal displacement is five left, and my vertical displacement is three down.

Therefore, vector negative five, negative three is the correct answer.

We can see Sofia is right.

What mistake has Andeep made? Andeep has not considered the direction of the vector.

Andeep started with the point at the bottom left of the line, but we can see that that arrow is clearly pointing from the top right to the bottom left.

It's really important that you check the direction of your vector.

Your turn now then to test that out.

Which is the correct graphical representation of this vector? And the vector is negative one, six.

I'd like you to decide whether you think that is A, B, C, or D.

And in addition to that, I'd like you to write down the correct column vector for the ones that aren't right.

So you'll have three column vectors correctly written, and one you'll have as the correct answer to the column vector, negative one, six.

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

Well done.

What was the correct answer? The correct answer was B.

B, we can see, look at where the arrow is.

The arrow is starting at the bottom right corner and moving to the top left.

So we need to be moving one place to the left and six places up.

That's the vector, negative one, six.

The vector for A is negative one, negative six, for C is one, six, and for D is one, negative six.

Really important that we check carefully which way the arrow is pointing.

Now for task A, I'd like you please to match each vector to its graphical representation.

There is one vector I've given you there, given you 10 diagrams, but I've given you 11 vectors.

One of them doesn't have a pair.

I'd like you to identify also which that one is.

So it'll be the one that's left at the end, won't it? Pause the video, match them up, check your counting carefully, and then when you are ready, come back and we'll check those answers for you.

Well done.

And now then we're ready for question 11.

I'd like you please to find the value of A in the following: We've got here the column vector is three and then a, and it represents that geographical representation.

What is the value of a? So you need to do that for each of those three diagrams please.

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

Great work.

Check those answers.

Number one was j.

Number two was b, number three was c, four was h, five was e, six was d, seven was i, eight was k, nine was g, and 10 was a.

How did you get on? Of course you got 10 outta 10, and you should have found that f was the one with no pair, the vector zero, four.

And well done if you decided that you were going to draw the matching pair for that one.

And then onto question 11, we were finding the value of a.

We look at the first one, we can see that it's moved three right and four up.

So a is four.

If we look at the B, then it's a, four.

The missing value is the horizontal displacement.

We can see it's moved two to the left, so that's negative two.

And for part C, here we are looking for that vertical displacement and we can see that actually it is not moved at all vertically.

So therefore a is zero.

Well done on that first learning cycle.

Now let's get started on the next one.

So we are going to be drawing this time, so we're gonna be drawing those column vectors.

So get your pencils and rulers at the ready.

I want some nice straight lines here please.

We're going to draw the vector four, six from A to B.

I know you can tell me what movement is the vector describing.

Yes, well done.

It's saying move four units right and six units up.

Let's do that.

Four units to the right and six units up.

There is point B.

Remember we were travelling from A to B, so our arrow must be pointing from A to B.

Let's take a look at another one.

Draw the vector, negative two, five, and again from A to B.

What movement is the vector describing this time? It's asking us to move two units to the left and five units up.

Let's do that.

Two units to the left and five units up.

That's where point B is going to be.

I remember we are moving from A to B.

Really important, we don't always assume that we are moving in alphabetical order.

We could have been moving from B to A.

And another one.

What movement is the vector describing? That's right, it's telling us to move three places to the right and one place down or units, I should really say.

I'm gonna move three units to the right, one unit down.

This is where point B is going to be and remember to draw on that arrow.

Sofia says, "Vectors are always diagonal lines." I'd like you please to write a sentence to help Sofia see why she's incorrect.

So pause the video, write me a sentence.

Remember, start with a capital letter, finish with a full stop, and then when you've got your answer, come back and we'll see whether you agree with my answer.

Good luck with this one.

Okay, wondering what you've written.

Hopefully you've written something like this.

Vectors can be used to describe solely horizontal or vertical displacement, so therefore we could just see a horizontal line or a vertical line.

They don't always represent just diagonal lines.

Write down a vector that describes solely horizontal displacement and also please a vector that describes solely vertical displacement.

Again, pause the video, write down your two vectors, one for solely horizontal displacement and one for solely vertical displacement.

When you've got your answers, come back and we'll check those for you.

Great work.

A zero is used in the column vector to represent no displacement horizontally or vertically.

Any vector of the form a at the top, zero at the bottom of the column vector describes solely a horizontal displacement.

So long as you've got a zero at the bottom and you've got a number at the top, then your answer will be correct.

Remember, a could be negative.

Any vector of the form zero, a describes solely vertical displacement.

Again here, as long as you've got zero at the top and then any value for a, that will represent a solely vertical displacement.

How did you get on? Superb.

I knew you'd get those right.

Draw the vector, negative five, zero from A to B.

What movement is the vector describing? It's saying five units left and zero up or down.

So zero vertically.

I'm just going to move five to the left, label the point B, and then make sure I've got my arrow pointing in the correct direction.

I'd like you please now to sort the following vectors into ones that are horizontal lines, ones that are vertical lines, and ones that are diagonal lines.

Pause the video, decide what goes in box and then when you're ready, come back and we'll check those.

Super work, well done.

Six, one is diagonal, it's moving vertically and horizontally, so it's going to be diagonal.

Zero, six is vertical.

We can see that the top number, the horizontal displacement is zero.

We're not moving any way to the left or to the right, but we are moving six up, so it's going to be a vertical line.

Six, zero, that's going to be horizontal.

The top number is saying we are moving six right, but we're not moving up or down.

So therefore it's a horizontal line.

Zero, zero, hmm, interesting one.

What do you think about that? What does that represent? It represents no movement at all.

I've moved zero places to the left or right and zero places vertically.

One, six, where's that going? And of course, you've put that in diagonal.

Negative six, zero is going to be horizontal.

There's only a horizontal displacement represented by the negative six, the vertical displacement is zero.

Negative one, six is diagonal, it has both vertical and horizontal displacement.

And then finally, zero, negative six was vertical.

The top number is zero, showing zero displacement horizontally, and the bottom number is negative six, indicating we're going to be moving six down.

So this will be a vertical line.

Draw the vector, negative five, four.

Now Izzy says, "I have only drawn vectors when I have a starting point.

How am I going to draw this vector if I do not know where it starts?" Sam says, "Remember, Izzy, vectors have a magnitude, so size, the length of the line, and a direction, so which way we are moving.

If a starting point is not given, you can choose your own starting point." Izzy's response is, "Of course, that makes sense.

I'll start by putting my point in the middle of the grid." And Sam says, "That is not going to work, Izzy." Oh, dear, Sam doesn't think that's going to work.

Why does Sam say that? And the reason Sam says that is the vector describes moving five units to the left and four units up, moving five to the left will be off the grid.

Where is the best place to position the starting point? If you're moving left and up, the best place is in the bottom right-hand corner.

Let's move our point to the bottom right-hand corner.

Now we can move five places to the left and four places up.

That's where my point is going to be, remembering my arrow to show the direction, my starting point to my end point.

Draw the vector five, negative three.

Where is the best place to position the starting point this time? If you're moving right and down, then the best place is in the top left-hand corner.

Let's place our point, our starting point in the top left corner, and then I'm moving five places to the right and three places down.

And then remember to join those with a nice straight-ruled line and that arrowhead.

Your turn now then to decide, for each of these, where is it best to put your starting point? Pause the video and then when you've got your six answers, come back.

Super well done.

Negative seven, negative five would be the top right because you are moving to the left and down.

Two, negative six would be top left, you are moving right and down.

Negative six, three would be the bottom right.

Two, eight would be bottom left.

Negative six, negative two is going to be top right, and four, negative three is going to be top left.

How did you get on? Well done.

Now you are ready to have a go at task B.

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

So you're gonna start with the vector three, one.

Now it's really important here that you don't draw on your working arrows because that's gonna get very, very messy very, very quickly.

So count three to the right and one up and just mark that with a point and then draw your vector on.

Each start, sorry, each end point is the starting point of the next vector.

Once you've done that, you should reveal a picture.

Pause the video, and good luck with this.

When you get back, we'll compare my picture to your picture and see that they're the same.

Of course, they'll be the same.

Great work.

And your picture is of what? It's of what? An elephant? Yes, the picture was an elephant.

So well done if your elephant looks identical to mine, that means that you've made no errors.

Fantastic and well done.

Now let's summarise the learning of today's lesson.

A vector can be used to describe a translation, remembering that the top number is our horizontal displacement and the bottom number is our vertical displacement.

If we are moving left, it's going to be negative, right, positive, and if we're moving down, negative, and up, positive.

A vector has magnitude, and that's the size, and it also has a direction.

The length of the line shows its magnitude, and the arrowhead points in the direction.

We can see here that the vector represented by the green arrow, the magnitude is the length of that line and the direction is which way the arrow is pointing, where is our starting point and where's our end point.

If no starting point is given, this can be anywhere on the grid.

But remember, you will need to ensure that you have enough space to draw your vector.

This is one of those times when it is really, really important that you use a pencil so that you can rub it out if you find out that you've run out of space on your grid.

Don't forget as well when you're drawing your vectors, a nice ruled line with that arrowhead.

Great work today.

I hope you enjoyed drawing your elephant.

I look forward to seeing you again really soon.

Do please take care of yourself.

I look forward to seeing you again really soon.

Bye.