# Question Video: Finding the Scalar Multiplied by a Vector Using a Given Figure Mathematics

Given the information in the diagram below, if ππ = πππ, find π.

02:35

### Video Transcript

Given the information in the diagram below, if the vector ππ is equal to π times the vector ππ, find π.

In our diagram, weβve actually been given two similar triangles. That is, one is an enlargement of the other. Triangle πππ has been enlarged onto πππ. Now, we know this since angle π΄ here is a shared angle. We see that angle π΄πΈπ· and π΄πΆπ΅ are equal as are angles π΄π·πΈ and π΄π΅πΆ. And this is because corresponding angles are equal and we know sides πΈπ· and πΆπ΅ are parallel.

Since the triangles have equal angles, they must be similar. And this means weβre 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΅πΆ and the original triangle to be π΄π·πΈ, 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. Itβs 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·πΈ onto π΄π΅πΆ, weβd multiply any of its lengths by the scale factor of five over two. Now, weβre trying to find a relationship between the vectors ππ and ππ. Well, we know that the dimension πΆπ΅ must be five over two times the dimension πΈπ·. But since these lines are also parallel, we know that they have the same direction. And this means we can say that the vector ππ must be five over two times the vector ππ.

The problem is, we want to work out the vector ππ in terms of the vector ππ. Currently, we have the vector ππ in terms of ππ. And so weβre traveling in the opposite direction. And we remember that to do this with vectors, we change the sign so that the vector ππ is equal to the negative vector ππ. We just showed that the vector ππ is five over two times the vector ππ. So this must mean that the vector ππ is negative five over two times the vector ππ. Comparing this to the original form in the question, and we see then that π is equal to negative five over two.