Lesson video

Hi, everyone.

My name is Ms. Ku.

And I'm really happy that you're joining me today.

Today, we'll be looking at vectors, a great topic.

And chances are, you've already used vectors today already, whether that be playing a game or using satellite navigation.

I hope you enjoyed today's lesson.

So let's start.

Hi, everyone and welcome to this lesson on the sum and difference with algebraic vector notation under the Unit: Vectors.

And by the end of the lesson, you'll be able to find the sum and difference with vectors written algebraically.

So let's have a look at some keywords.

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

Today's lesson will be broken into two parts.

We'll be looking at adding and subtracting vectors first, and then moving on to adding and subtracting vectors from diagrams. So let's make a start, adding and subtracting vectors.

Here we have two vectors.

I want you to fill in the column vectors for a and b.

See if you can give it a go.

Press pause if you need.

Great work.

Let's see how you got on.

Well, a has the column vector 1, 3 and b has the column vector of 2, -3.

Well done if you got this.

Now, Andeep puts two of vector a together.

How could we simplify this? You can use the grid to help.

Well, it would be simply 2a written algebraically or in column form it'd be 2, 6.

So farther than writing it as a and a on our grid, we can write it on our square grid as 2a.

Now, Andeep puts two of vector a together and then puts three of vector b together.

How do you think we could simplify three lots of our vector b? Well, it'd be simply 3b or in its column form 6, -9.

Now, let's have a look at that grid, b and a b, and a b.

Well, we definitely can simplify it to be 3b and it looks like this on our square grid.

So using the grid, what do you think the column vector is of 2a + 3b? And you can see it here.

Well, looking at our horizontal component first, it's simply 8 and then our vertical component is -3.

So the column vector of 2a + 3b is 8, -3, but without the grid, how else do you think we can work out the column vector form of 2a + 3b? Well, 2a + 3b means two lots of vector a add three lots of vector b.

So if we add those horizontal components, we have 8.

If we add those vertical components, we have -3.

Well done if you spotted this.

So the resultant horizontal component is the sum of the horizontal components and the resultant vertical component is the sum of the vertical components.

For example, we have c = in column form 4, 5 and d = -1, 3.

And we're asked to work out the vector 3c + 2d.

Well, we have 3 as a multiplier of c and we have 2 as a multiplier of d.

So let's use our priority of operations.

We multiply first.

Using multiplication, we can see 3 multiplied by a column vector c gives us 12, 15, and 2 multiplied by a column vector of d, gives us -2, 6.

From here, summing those horizontal components and summing those vertical components gives us the vector 3c + 2d to be 10, 21.

Really well done if you've got this.

Now, it's time for a check.

Given these vectors, p is 2, -1 and q is 3, 5 and r is -4, 7, I want you to work out the vector 3p + 2q, work out the vector 4p + 2r, and work out the vector 2p + 3q + 0.

5r.

See if you can give it a go.

Press pause if we need more time.

Well done.

Let's see how you got on.

Well, 3p + 2q means we have three lots of our vector p add two lots of our vector q.

So writing it down, we have 3 multiply by our 2, -1, add 2 multiply by our 3, 5.

This gives us 6, -3 add 6, 10.

Summing those horizontal components and summing those vertical components gives me a column vector of 12, 7.

So therefore 3p + 2q is simply 12, 7.

Next, we have 4p + 2r This means 4 lots of vector p add 2 lots of vector r.

So multiplying 4 by our vector p gives us 8, -4 and multiplying our 2 by the vector r gives us -8, 14.

Summing those horizontal components and summing those vertical components gives us the final vector of 4p + 2r is 0, 10.

Really well done if you've got this.

For c, well, we have 2 lots of our vector p add 3 lots of our vector q at 0.

5 times our vector r, using our knowledge and priority operations.

And then summing all those horizontal components and summing all those vertical components gives us the final vector to be 11, 16.

5.

Really well done if you got this.

Now, what I want you to do is have a look at this diagram and I want you to see if you can explain how 2a - 3b is formed from our grid? Given the fact that we know a in column form is 1, 4 and vector b is 2, 3.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, 2a in column vector form is 2, 8 and 2a is shown as this vector.

You can see my direction and I've labelled it algebraically, 3b has the column vector form 6, 9.

All I'm doing here is showing 3b direction and the algebraic form here.

So the resultant vector is formed by 2a and then the negative of 3b.

So we want to travel in the opposite direction of 3b.

So that means 2a - 3b is in column vector form -4, -1.

Really well done if you got this.

So removing the grid, how can we work out the resultant vector? Well, the resultant vector is made by summing 2a and -3b.

So we can identify our multipliers.

2 multiplied by our vector a, add -3, multiply by our vector b.

Using our priority of operations, we multiply first, then we sum giving me the final answer of -4, -1.

Well done if you've got this.

Now, it's time for your check.

Given these vectors, I want you to work out the answer in column form.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for a, we should have got -2, -12.

For b, we should have had 18, -19.

And for c, big questions.

So well done if you got this right, 26, -15.

Really well done.

Sofia says, "We've used algebraic vectors to write the column vector, but how do we use the column vector to write the algebraic vector?" For example, c in column form is 2, 5 and d in column vector form is 2, -1, and we're asked to write the resultant in terms of c and d.

And Aisha says, "There are quite a few different ways here, but finding the multiplier is the easiest." So let's have a look at this column vector.

The column vector here is the vector 4, 10, therefore, this must be 2c as we have a scale and multiplier two.

So we know 2c is the vector 4, 10.

I'm gonna label it here.

Let's have a look at our other vector.

Well, this vector in column vector form is 8, -4.

So let's say we can identify the multiplier.

Well, the multiplier here must be 4, so therefore the vector must be 4d.

In other words, we multiply our column vector d by 4, giving us the vector 8, -4.

I'm going to label it here.

Therefore, we now know the resultant vector is 2c + 4d.

Now, it's time for your check, given that a as a column vector is 3, 4, and b as a column vector is 2, -2, we're asked to write the resultant vector in terms of a and b.

See if you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, we know vector a is 3, 4 and b is 2, -2.

So let's identify this vector here at 6, 8.

If it's 6, 8, I'm simply multiplying our vector a by 2.

So this has to be vector 2a.

Looking at our next vector, you can see it has the column vector form of 4, -4.

So therefore I'm multiplying vector b by 2.

So I'm going to label this vector as 2b.

So what's our resultant vector in terms of a and b? It's simply 2a + 2b.

Great work, everybody.

Now, it's time for your task.

I want you to work out the following in column form.

Given we know a is 2, 3, b is -5, -2, c is -3, -1 and d is -4, 0.

See if you can give it a go.

Press pause if you need more time.

Great work, everybody.

Let's move on to question two.

Given these vectors a, which is 2, 3, b which is -5, -2, and c, which is -3, -1.

On the grid, draw the resultant vectors.

See if you can give it a go.

Press pause for more time.

Well done.

Let's move on to question three.

Given these vectors a is 1, 3, b is 2, -3 and c is 0, 1, I want you to write the following in terms of a, b and/or c.

See if you can give it a go.

Press pause for more time.

Question four, one of my favourite questions.

It's vector golf.

Get the ball in the hole in as few shots as possible.

Any multiple of a vector is classed as a shot.

For example, 3a + 2b counts as two shots as 3a is one shot and 2b is another shot.

See if you can give it a go.

Press pause for more time.

Well done.

Let's move on to Hole 3 and 4, same applies.

See if you can give it a go.

Press pause for more time.

Great work, everybody.

Let's go through these answers.

Well for question one, here's all my wonderful working out and the final answers.

Press pause if you need more time to mark.

For question two, here are my drawings.

Remember a vector can exist anywhere as long as the magnitude or the length and the direction is the same.

You have these correct if you have these vectors.

Really well done if you got this right.

For question three.

We should have these following answers.

Mark them.

Press pause if you need more time.

For question four, there's lots of different ways.

I'm just gonna give you some examples.

So for Hole 1, I managed to do it in two shots.

For Hole 2, same.

Again, I managed to do it in two shots.

For Hole 3, I managed to do it in two shot and for Hole 4, this was a tricky one.

It took me three shots here.

Did you do better? If you did? Really well done.

Great work, everybody.

So now it's time for the second part of our lesson.

Adding and subtracting vectors from diagrams. So far we know the resultant horizontal component is the sum of the horizontal components and the resultant vertical component is the sum of the vertical components.

For example, when a is 1, 3 and b is 2, -3, therefore, 3a + 2b is simply going to be 7, 3.

And you can see my working out here.

We also know that we can represent column vectors algebraically.

For example, you can see our resultant vector is 3a + 2b.

Therefore, we can move away from using column vectors and keep to algebraic terms. And forming vector pathways allows a structured approach when finding vectors and resulting vectors.

For example, here's a quadrilateral, A, B, C, and D.

We know the vector AB is 2a and we know the vector BC is 3b and a.

And we also know the vector CD is equal to -a.

And the question wants us to find AC in terms of a and b.

Firstly, label the algebraic vectors if they're not already on the diagram and ensure that you put the direction.

So let's label, we know the vector A to B is 2a, the vector B to C is three 3b and a and the vector C to D is -a.

Now, we're going to write our vector path starting from A and finishing at C using our known vectors.

So while vector AC is the resultant vector of AB, add the vector BC.

So let's see if we can sum these together.

We're going to substitute the algebraic terms for each vector.

We know vector AB is 2a and we know the vector BC is 3b and a.

Summing these together and simplifying to find in terms of a and b, we now know vector AC is 3a + 3b.

Now, what I want you to do is a check, which of the following is the correct vector for XY? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, it should be d, 4b and 5a.

Basically, we are summing those like terms. Very well done if you got this.

Now, it's also important to remember the impact when reversing vectors.

For example, here we have a quadrilateral, OXZ and Y.

We know OY is 5b, OX is 7a and the vector YZ is 5a - b.

And we're asked to identify what's the vector XY in terms of a and b? Well, first of all, just like always, we always label the vectors if they're not given the diagram, ensuring we put the direction too.

So we know OY is 5b, we know OX is 7a, and we know YZ is 5a - b.

Now, we're going to write our vector path starting from X and finishing at Y using these known vectors.

So we know the vector XY is the resultant to XO + OY.

XO has to be -7a.

As we know OX is 7a.

So I'm going to substitute this in X to O is -7a and O to Y is our 5b.

Therefore, finally writing our vector of XY as -7a + 5b.

I just want to stress that both of these vectors are exactly the same.

Some people tend to write the positive term first, followed by a negative term, but both are exactly the same.

Well done.

Now, it's time for your check using the quadrilateral OXZY.

I want you to write the vector ZO in terms of a and b.

See if you can give it a go.

Press pause for more time.

Great work.

Let's see how you got on.

Well, we know the resultant vector of ZY + YO is ZO.

So let's substitute.

Well, we know the vector ZY is -5a - b, and we're adding the vector YO, which is -5b.

So expanding out our brackets, we have -5a + b - 5b gives us -5a - 4b or you could rewrite it as -4b - 5a.

Really well done if you got this.

Now, it's time for another check.

I want you to work out the following vectors in terms of a, b, c, and/or d.

See if you give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, for the vector OB, we can sum vector OA and AB, giving us a final vector of 3a - b.

For vector BE, we can sum, vector B to C, add the vector C to D, add the vector D to E, which gives us this.

And in simplified form, -a - 4b + d.

Well done if you've got this.

Great work, everybody.

Now, it's time for your task.

I want you to work out the following in terms of p and a.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

Work out the following vectors in terms of a, b, c and/or d.

See if you can give it a go.

Press pause for more time.

Well done.

For question three, I want you to write the following vectors in terms of a and b.

Give it a go.

Press pause for more time.

Great work.

Question four, this is a fantastic question.

Explain what is wrong with the following and correct each diagram.

OPQR is a quadrilateral and OPQR is a parallelogram.

See if you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, here are our answers to question one.

Question two, here's our answers.

Really well done if you've got this.

And for question three, labelling on our diagram makes it easier to identify our answers.

Very well done if you've got this.

And for question four, well, the resultant vector PQ should be equal to 12b - 6a.

You can either amend the vector PQ to make it correct or you could amend the vector PO.

Really well done if you got this one.

Next, we know the resultant vector PQ should it be equal to -5b - a.

We can amend the vector PQ to make it correct or we could amend the other vectors.

Really well done if you got this.

Great work, everybody.

So in summary, the resultant horizontal component is the sum of the horizontal components and the resultant vertical component is the sum of the vertical components.

For example, when c is 4, 5 and d is -1, 3, and the question wants us to work out 3c + 2d, we simply multiply our vector c by 3 and sum it two, to multiply by our vector d.

This gives us the vector in column form to be 10, 21.

We can also represent column vectors algebraically, therefore, can move away from using column vectors and keep to algebraic terms where the resultant vector can be found by summing these like terms. Great work, everybody.

It was wonderful learning with you.