### Video Transcript

The angle between vector π and
vector π is 22 degrees. If the magnitude of vector π is
equal to three times the magnitude of vector π is equal to 25.2, find the dot
product between π and π to the nearest hundredth.

In this question, weβre given some
information about two vectors π and π. First, weβre told the angle between
these two vectors is equal to 22 degrees. Next, weβre also told information
about their magnitudes. We know the magnitude of π is
equal to 25.2, and we know that three times the magnitude of π is also equal to
25.2. So the magnitude of π is three
times bigger than the magnitude of π. We need to use this to find the dot
product of π and π. And we need to give our answer to
the nearest hundredth.

To answer this question, we need to
notice that we know a formula which connects the angle between two vectors with
their dot product. We recall if π is the angle
between two vectors π and π, then we know that the cos of π must be equal to the
dot product between π and π divided by the magnitude of π times the magnitude of
π. And in this question, we already
know some of these values. For example, weβre told the angle
between our two vectors is 22 degrees. Next, weβre also told the magnitude
of π is equal to 25.2.

And we could also find the
magnitude of π using the information given to us in the question. One way of doing this is to notice
that three times the magnitude of π is equal to 25.2. We can then solve this to find the
magnitude of π by dividing both sides of our equation through by three. And calculating this, we get the
magnitude of π is 8.4. So, in fact, the only unknown in
this equation is the dot product between π and π. And thatβs exactly what weβre asked
to calculate.

So weβll substitute the angle of π
equal to 22 degrees, the magnitude of π equal to 25.2, and the magnitude of π
equal to 8.4 into our equation. This gives us the cos of 22 degrees
should be equal to the dot product between π and π divided by 25.2 times 8.4. And now we can just rearrange this
equation for the dot product between π and π. We multiply through by 25.2 times
8.4. This gives us π dot π is 25.2
times 8.4 times the cos of 22 degrees. And we can calculate this to the
nearest hundredth or to two decimal places. Itβs equal to 196.27.