Question Video: Finding the Dot Product between Vectors of an Equilateral Triangle | Nagwa Question Video: Finding the Dot Product between Vectors of an Equilateral Triangle | Nagwa

Question Video: Finding the Dot Product between Vectors of an Equilateral Triangle Mathematics • Third Year of Secondary School

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If ๐ด๐ต๐ถ is an equilateral triangle of side length 29.9 cm, find ๐€๐ โ‹… (๐€๐‚ + ๐‚๐).

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Video Transcript

If ๐ด๐ต๐ถ is an equilateral triangle of side length 29.9 centimeters, find the dot product between the vector ๐€๐ and the vector ๐€๐‚ plus ๐‚๐.

In this question, weโ€™re given some information about an equilateral triangle ๐ด๐ต๐ถ. Weโ€™re 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 ๐€๐ and the sum of two vectors, ๐€๐‚ plus ๐‚๐. All three of these vectors are side lengths of our equilateral triangle.

In fact, thereโ€™s several different ways we can evaluate this expression, and weโ€™ll go through a few of these. The first thing weโ€™re going to do, however, is sketch a picture of our equilateral triangle. So, letโ€™s start with an equilateral triangle ๐ด๐ต๐ถ, 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 ๐€๐‚ plus the vector ๐‚๐. The vector ๐€๐‚ starts at the point ๐ด and ends at the point ๐ถ, but then the vector ๐‚๐ starts at the point ๐ถ and ends at the point ๐ต. So, in fact, this entire vector is just going to be the vector from ๐ด to ๐ต, since it starts at ๐ด and ends at ๐ต.

And it might be easier to see this on our diagram. First, we can start with our vector from ๐ด to ๐ถ. Then, we can add on our vector from ๐ถ to ๐ต. And we can see this is exactly the same as the vector from ๐ด to ๐ต. And in fact, weโ€™ll be able to use this to directly evaluate the dot product given to us in the question. By replacing the vector ๐€๐‚ plus the vector ๐‚๐ with the vector ๐€๐, all we need to do is evaluate the dot product between the vector ๐€๐ and itself.

And in fact, thereโ€™s 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 ๐ฏ and itself is equal to the magnitude of the vector ๐ฏ squared. And we actually know the magnitude of the vector ๐€๐ since itโ€™s 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 isnโ€™t the only way we could have answered this question. We could have also done this by using properties of the dot product. Weโ€™ll start with the dot product weโ€™re asked to calculate. And weโ€™ll notice something interesting. Weโ€™re taking the dot product between the vector and the sum of two vectors. And thereโ€™s a useful property we know about situations like this. The dot product distributes over vector summation. In other words, for any three vectors ๐ฎ, ๐ฏ, and ๐ฐ, the dot product between the vector ๐ฎ and ๐ฏ plus ๐ฐ will be equal to ๐ฎ dot ๐ฏ plus ๐ฎ dot ๐ฐ. 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 weโ€™re asked to calculate, we get the dot product between the vector ๐€๐ and the vector ๐€๐‚ and then we add the product between the vector ๐€๐ and the vector ๐‚๐. 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œƒ is the angle between two vectors ๐ฎ and ๐ฏ, then the cos of ๐œƒ will be equal to the dot product between ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ. We can rearrange this formula by multiplying both sides through by the magnitude of ๐ฎ times the magnitude of ๐ฏ, giving us the dot product between two vectors ๐ฎ and ๐ฏ is equal to the magnitude of ๐ฎ times the magnitude of ๐ฏ multiplied by the cos of the angle between the two vectors ๐ฎ and ๐ฏ.

We can apply this to evaluate both of the dot products weโ€™re 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 ๐€๐ and the vector ๐€๐‚ is 60 degrees. In fact, exactly the same is true about the vectors ๐€๐ and ๐‚๐. 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 ๐‚๐ which starts at ๐ด. 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 ๐ด๐ต๐ถ is an equilateral triangle of side length 29.9 centimeters, then thereโ€™s a lot of different methods we could use to find the dot product between the vector ๐€๐ and the vector ๐€๐‚ plus the vector ๐‚๐. In all of these cases, we would get the answer of 894.01.

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