### Video Transcript

The figure shows two vectors, ๐ฏ
and ๐ฎ, where the magnitude of ๐ฏ is equal to five and the magnitude of ๐ฎ is equal
to seven. Use the parallelogram method to
find the magnitude of the resultant of these two vectors. Give your answer to two decimal
places.

The parallelogram method is so
called because of the shape that it creates, as shown in the figure. This says that if two vectors
acting simultaneously at a point can be represented both in magnitude and direction
by the adjacent sides of a parallelogram drawn from a point. Then the resultant vector is
represented both in magnitude and direction by the diagonal of the parallelogram
passing through that point.

So as the diagram shows, the
resultant of ๐ฏ and ๐ฎ โ thatโs their sum โ is given by the diagonal of the
parallelogram thatโs drawn in pink. But how does that help? Well, weโre given some information
about the magnitude of the vectors ๐ฎ and ๐ฏ. Remember, the magnitude is the size
or the length of the vector. So we can label ๐ฎ as seven units
and ๐ฏ as five units. We can also add in some missing
angles. We know that cointerior angles sum
to 180 degrees. And we know that these sides are
parallel. Itโs a parallelogram. So this angle here is given by 180
minus 125, which is equal to 55 degrees.

Now, in fact, we can label the
lengths of two further sides in our parallelogram. We know that opposite sides in a
parallelogram are equal in length. So we can label these sides as five
and seven. And now weโre going to split our
parallelogram into two triangles. Weโre trying to find the magnitude
of ๐ฏ plus ๐ฎ, so the length of that diagonal line.

Letโs call that length ๐ฅ
units. We now see we have a
non-right-angled triangle, for which we know the lengths of two of its sides and the
angle between them. This means we can use the cosine
rule to find the missing length ๐ฅ. The cosine rule says that ๐
squared is equal to ๐ squared plus ๐ squared minus two ๐๐ cos ๐ด. Since the angle we have is 55
degrees, we label this vertex ๐ด. Then the side opposite is lowercase
๐. We can label the other two sides in
any order. Letโs label five as ๐ and seven as
๐.

We can substitute everything we
know about our triangle into this formula. And we get ๐ฅ squared equals five
squared plus seven squared minus two times five times seven times cos of 55
degrees. Evaluating this expression on the
right-hand side gives us 33.84 and so on. Now, weโll solve this equation for
๐ฅ by taking the square root of both sides. That gives us 5.818 and so on,
which, correct to two decimal places, is 5.82. So we found the length of ๐ฅ to be
5.82 units, which means the magnitude of the resultant of ๐ฎ and ๐ฏ is 5.82.