Video Transcript
If ๐ด๐ต๐ถ is an equilateral triangle of side length 29.9 centimeters, find the dot product between the vector ๐๐ and the vector ๐๐ plus ๐๐.
In this question, weโre given some information about an equilateral triangle ๐ด๐ต๐ถ. Weโre told that the side length of this equilateral triangle is 29.9 centimeters. We need to use this information to determine the value of the dot product between the vector ๐๐ and the sum of two vectors, ๐๐ plus ๐๐. All three of these vectors are side lengths of our equilateral triangle.
In fact, thereโs several different ways we can evaluate this expression, and weโll go through a few of these. The first thing weโre going to do, however, is sketch a picture of our equilateral triangle. So, letโs start with an equilateral triangle ๐ด๐ต๐ถ, where we know all of the side lengths are 29.9 centimeters. And also because this is an equilateral triangle, we also know all of the internal angles are going to be 60 degrees.
This then gives us several different methods to evaluate this dot product. The easiest way will be to notice something interesting about the vector ๐๐ plus the vector ๐๐. The vector ๐๐ starts at the point ๐ด and ends at the point ๐ถ, but then the vector ๐๐ starts at the point ๐ถ and ends at the point ๐ต. So, in fact, this entire vector is just going to be the vector from ๐ด to ๐ต, since it starts at ๐ด and ends at ๐ต.
And it might be easier to see this on our diagram. First, we can start with our vector from ๐ด to ๐ถ. Then, we can add on our vector from ๐ถ to ๐ต. And we can see this is exactly the same as the vector from ๐ด to ๐ต. And in fact, weโll be able to use this to directly evaluate the dot product given to us in the question. By replacing the vector ๐๐ plus the vector ๐๐ with the vector ๐๐, all we need to do is evaluate the dot product between the vector ๐๐ and itself.
And in fact, thereโs a lot of different ways we can evaluate just this dot product. The easiest way is to recall the following fact: the dot product between any vector ๐ฏ and itself is equal to the magnitude of the vector ๐ฏ squared. And we actually know the magnitude of the vector ๐๐ since itโs the side of an equilateral triangle where every side is 29.9 centimeters. Therefore, this is just equal to 29.9 squared. And we can calculate this; it gives us our final answer of 894.01.
However, this isnโt the only way we could have answered this question. We could have also done this by using properties of the dot product. Weโll start with the dot product weโre asked to calculate. And weโll notice something interesting. Weโre taking the dot product between the vector and the sum of two vectors. And thereโs a useful property we know about situations like this. The dot product distributes over vector summation. In other words, for any three vectors ๐ฎ, ๐ฏ, and ๐ฐ, the dot product between the vector ๐ฎ and ๐ฏ plus ๐ฐ will be equal to ๐ฎ dot ๐ฏ plus ๐ฎ dot ๐ฐ. Instead of taking the dot product of the sum of our two vectors, we can instead take the dot product of each separately and add the results.
So, by applying this to the dot product weโre asked to calculate, we get the dot product between the vector ๐๐ and the vector ๐๐ and then we add the product between the vector ๐๐ and the vector ๐๐. And we have a few different options for calculating the dot product between these vectors. One way we could do this is to find component-wise definitions of each of these vectors. We would do this by using trigonometry on our equilateral triangle, and this would work. However, we also know the angle between all of these vectors. And we know a formula which involves the dot product and the angle between two vectors.
We recall if ๐ is the angle between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ will be equal to the dot product between ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ. We can rearrange this formula by multiplying both sides through by the magnitude of ๐ฎ times the magnitude of ๐ฏ, giving us the dot product between two vectors ๐ฎ and ๐ฏ is equal to the magnitude of ๐ฎ times the magnitude of ๐ฏ multiplied by the cos of the angle between the two vectors ๐ฎ and ๐ฏ.
We can apply this to evaluate both of the dot products weโre left with in our expression. First, we need to notice all three of the vectors given to us in this expression are side lengths of our triangle. So, all three of these have magnitude 29.9. Next, we can also see in our diagram the angle between the vector ๐๐ and the vector ๐๐ is 60 degrees. In fact, exactly the same is true about the vectors ๐๐ and ๐๐. However, we do need to be a little bit more careful. To find the angle between these two vectors, we need to construct a vector equivalent to the vector ๐๐ which starts at ๐ด. Then, we need to use a little bit of geometry to show that the angle between these two vectors is also equal to 60.
Now, we can just use our formula to evaluate all of these dot products. The magnitude of all of these vectors is 29.9. And the angle between both of these pairs of vectors is 60 degrees. And of course, we know the cos of 60 degrees is one-half. So, this gives us 29.9 times 29.9 multiplied by one-half plus 29.9 times 29.9 multiplied by one-half. And this gives us the same answer of 894.01; however, this method was more difficult than the previous one.
Therefore, we were able to show if ๐ด๐ต๐ถ is an equilateral triangle of side length 29.9 centimeters, then thereโs a lot of different methods we could use to find the dot product between the vector ๐๐ and the vector ๐๐ plus the vector ๐๐. In all of these cases, we would get the answer of 894.01.
