Lesson video

Hi, I'm Mr. Bond, and in this lesson, we're going to learn how to prove that two vectors are parallel.

We'll start by thinking about the condition that makes two vectors parallel.

Two vectors are parallel if one is a multiple of the other.

Let's take a look at some examples.

We'll use the vectors AB, CD, and EF.

Take a look at vector AB.

How could we write this as a column vector? AB is two, three written as a column vector.

What about CD? CD is four, six written as a column vector.

What do you notice? Vector CD is equal to vector AB multiplied by two.

The fact that it's multiplied by two means that CD is a multiple of AB.

So that means that AB is parallel to CD.

Now let's compare vector CD to vector EF.

We know that vector CD is given by the column vector four, six.

What about vector EF? How would we write that as a column vector? We'd write it as the column vector seven, nine.

So we'd have to multiply six by three over two to get nine.

Now, it isn't a problem that we're not multiplying it by an integer, but we must multiply the other number, four, by three over two if these vectors are parallel, but to go from four to seven, we'd multiply by seven over four.

So that means that EF is not a multiple of CD, and therefore, these two vectors are not parallel.

Here are some more vectors for you to think about.

Is the vector three, two parallel to the vector negative 15, negative 10? Pause the video to have a think and resume the video once you're finished.

Yes, it is.

Negative 15, negative 10 is equal to three, two multiplied by negative five.

We mustn't forget that there are other notations for vectors as well as column notation.

Here we have the vector a, we're going to compare this to two more vectors, the vector three a, and the vector four a plus b.

First, let's think about whether the first two vectors are parallel, a and three a.

Well, three a is three multiplied by a.

So three a is a multiple of a and these two are parallel.

What about a and four a plus b? Well, we can see that four a plus b is not just a multiple of a.

So therefore, these two vectors are not parallel.

What about three a and four a plus b? Well, again, we can't multiply three a by an integer to give us four a plus b.

So these vectors are still not parallel.

Here's a question for you to try.

Pause the video to complete your task and resume the video once you're finished.

Here are the answers.

In part a, it's true, they are parallel.

We know because 15, 21 is a multiple of the vector five, seven.

In part b, this is false, they're not parallel, because two a plus b is not a multiple of a plus b.

And in part c, it's true, they are parallel.

But what multiple of three c subtract d is three over two c subtract 1/2 d? Well, it's 1/2 multiplied by three c subtract d.

In this example, we're going to be thinking about the triangle ABC.

We can see that there's a point P on the side BC.

And P is the point such that the ratio BP to PC is equal to five to three.

So this means that the point P is 5/8 down that side going from B to C.

The first thing that I want to do is write the vector BC in terms of r and w.

So, looking at our triangle, I can see that the vector from B to C is equivalent to the sum of the vectors BA and AC.

So this is equal to negative five r add three w.

The next thing I'd like to do is show that the vector AP is parallel to r plus w.

Well, how would we do that? First, let's describe the vector AP in terms of the other vectors that we know.

So, to get from A to P, I could go from A to B and then 5/8 of the way from B to C.

So the vector AP is equal to the vector AB plus 5/8 of the vector BC.

We've just worked out what the vector BC is in terms of r and w, so let's substitute this into our equation.

So the vector AP is equal to five r, the vector AB, plus 5/8 negative five r plus three w.

If we expand the bracket, that gives us this.

Five r subtract 25/8 r add 15/8 w.

And then if we simplify, we'll get the vector AP is equal to 15/8 r add 15/8 w.

We can take out a factor.

Let's factorise this.

So now we know that the vector AP is equal to 15/8 r plus w.

This is a multiple of r plus w, and therefore, it's parallel.

Here's a question for you to try.

Pause the video to complete your task and resume the video once you're finished.

Here are the answers.

In the first part of part a, we know that the vector AC is equal to j plus k, and this is pretty clear from the diagram.

And then, for the vector MN, we know that M and N are midpoints of AD and CD respectively so that's how we know that it's 1/2 j plus 1/2 k.

And then, yes, we know that they're parallel, because Ac is equal to the vector MN multiplied by two.

Here's the final question for this lesson.

Pause the video to complete your task and resume the video once you've finished.

Here are the answers.

The first thing that we need to do is label the points E and F on our diagram, because they're not already given.

E is the midpoint of BC and F is the midpoint of AC.

We want to show that the vector EF is parallel to the vector DC.

So what is the vector DC? We know, from the diagram, that that's equal to three q.

Now we need to think about the vector EF.

The vector EF is equal to 1/2 vector BC add 1/2 vector CA.

So now we need to think what the vectors BC and CA are in terms of of q and p.

When we've worked this out, we can substitute this into our equation.

So the vector EF is equal to 1/2 multiplied by negative q subtract two p add three q add 1/2 multiplied by negative three q add two p.

Simplifying this gives that the vector EF is equal to 1/2 multiplied by two q subtract two p add 1/2 multiplied by negative three q add two p.

If we expand each of these brackets and simplify, we see that the vector EF is equal to negative q.

So that means that the vector EF is equal to 1/3 multiplied by the vector DC, and therefore, DC and EF are parallel.

That's all for this lesson.

Thanks for watching.