### Video Transcript

The two vectors ๐ฏ and ๐ฎ are shown in the diagram. The following parallelogram is drawn based on the two vectors ๐ฏ and ๐ฎ. Using the properties of parallelograms, find ๐.

So, the first thing we need to do if weโre dealing with a parallelogram is mark on our parallel sides. And that, you can see from the diagram we have two pairs of parallel sides. So, the first relationship that we can use is this one here. So, we can see that our 50 degrees and our ๐ have either side of them parallel lines. So therefore, we can use the relationship known as the supplementary angles relationship.

This tells us that supplementary angles sum to 180 degrees. So therefore, we can say that ๐ is gonna be equal to 180 minus 50. Thatโs because, as we said, they are supplementary angles. And as you can see, Iโve written that down next to the 180 minus 50. Thatโs because whenever youโre doing this type of question, you need to give reasoning when youโre trying to find your answers. So therefore, we can say the value of ๐ is gonna be 130 degrees. And Iโve also shown this now on our diagram. Well now, there is in fact a second part to the question. And now that weโve answered the first part, letโs move on to the second part.

So now, the second part of the question says that given that ๐ฏ equals 13 and ๐ฎ equals 11, so thatโs our vectors, use the cosine rule to find the magnitude of ๐ฏ plus ๐ฎ. Give your answer to two decimal places.

So the first thing we need to do is remember what the cosine rule tells us. And it tells us that if we have a triangle, ๐๐๐, then ๐ squared equals ๐ squared plus ๐ squared minus two cos ๐ด, where ๐ด is the angle opposite the side ๐. So now that we have the cosine rule, letโs apply it to our situation that weโve got in the second part of our question.

Well, we can apply the cosine rule to our scenario because weโve got a triangle, and weโve got our angle ๐ด, because thatโs our 130 degrees. And then weโve got ๐, the side, because this is the side opposite to 130 degrees, which is ๐ฏ plus ๐ฎ. And therefore, as we said that ๐ฏ plus ๐ฎ is equal to ๐, we can say that ๐ squared is gonna be equal to 11 squared plus 13 squared, because that was our ๐ฎ and our ๐ฏ, minus two multiplied by 11 multiplied by 13 cos of 130. And we got these values because our ๐ and ๐, or ๐ฎ and ๐ฏ, which were 11 and 13 have also shown that on our diagram, because Iโve put ๐ฎ on the right-hand side as well, because this is gonna be vector ๐ฎ as well as the left-hand side. So, what weโre gonna get is that ๐ squared is equal to 290 minus 286 cos 130.

So now what we need to do, we need to take the square root of both sides of the equation. And thatโs because weโre trying to find ๐ because ๐ is our ๐ฏ plus ๐ฎ. And if you square root ๐ squared, you just get ๐. And whatever you do to one side of the equation, you have to do to the other. And when we do that, we get ๐ is equal to 21.767, et cetera.

So, have we finished here? Well, no, because we want the answer rounded to two decimal places. And to round our answer to two decimal places, Iโve put a line after the second decimal place. Now, look to the digit to the right of it cause this is gonna be our deciding digit. And if this is five or above, it means weโre gonna round the six up to a seven. Therefore after rounding, and given that ๐ฏ equals 13 and ๐ฎ equals 11, weโve used the cosine rule to find the magnitude of ๐ฏ plus ๐ฎ. And, the answer is ๐ฏ plus ๐ฎ is equal to 21.77, and thatโs to two decimal places.