### Video Transcript

Given the information in the
diagram below, if the vector ππ is equal to π times the vector ππ, find π.

In our diagram, weβve actually been
given two similar triangles. That is, one is an enlargement of
the other. Triangle πππ has been enlarged
onto πππ. Now, we know this since angle π΄
here is a shared angle. We see that angle π΄πΈπ· and π΄πΆπ΅
are equal as are angles π΄π·πΈ and π΄π΅πΆ. And this is because corresponding
angles are equal and we know sides πΈπ· and πΆπ΅ are parallel.

Since the triangles have equal
angles, they must be similar. And this means weβre able to
calculate a scale factor of enlargement. This is found by dividing the
length of the enlarged triangle by the corresponding length of the original. We sometimes write this as new
length divided by old length.

If we take the enlarged triangle to
be π΄π΅πΆ and the original triangle to be π΄π·πΈ, we see that we can find the scale
factor by dividing the length π΄π΅ by the length π΄π·. The length of π΄π΅ is actually the
sum of the two dimensions given. Itβs 7.8 plus 5.2, which is 13
centimeters. And so the scale factor for
enlargement here is 13 divided by 5.2, which is five over two.

Now, the scale factor is simply a
multiplier. We know that to enlarge triangle
π΄π·πΈ onto π΄π΅πΆ, weβd multiply any of its lengths by the scale factor of five
over two. Now, weβre trying to find a
relationship between the vectors ππ and ππ. Well, we know that the dimension
πΆπ΅ must be five over two times the dimension πΈπ·. But since these lines are also
parallel, we know that they have the same direction. And this means we can say that the
vector ππ must be five over two times the vector ππ.

The problem is, we want to work out
the vector ππ in terms of the vector ππ. Currently, we have the vector ππ
in terms of ππ. And so weβre traveling in the
opposite direction. And we remember that to do this
with vectors, we change the sign so that the vector ππ is equal to the negative
vector ππ. We just showed that the vector ππ
is five over two times the vector ππ. So this must mean that the vector
ππ is negative five over two times the vector ππ. Comparing this to the original form
in the question, and we see then that π is equal to negative five over two.