If 𝐴𝐵𝐶 is an equilateral triangle of side 4.75, find the dot product between the vectors 𝐀𝐁 and 𝐀𝐂 approximated to the nearest hundredth.
In this question, we’re given some information about a triangle 𝐴𝐵𝐶. First, we’re told that this is an equilateral triangle. We’re also told that the length of the side of this triangle is going to be equal to 4.75. We need to use this information to determine the dot product between the vectors represented by two of the sides of this triangle. That’s the dot product between 𝐀𝐁 and 𝐀𝐂. And we need to give this answer to the nearest hundredth.
The first thing we’ll do is sketch our triangle 𝐴𝐵𝐶. First, the question tells us the lengths of our sides is going to be 4.75. And this is an equilateral triangle, so all of the side lengths will have this value. And we also know because this is an equilateral triangle, all of the internal angles are going to be equal to 60 degrees. And this is enough information to give us several different methods of working out the dot product between these two vectors.
The easiest way to do this will be to use our formula involving the angle between two vectors. And the reason we might want to do this is if we were to sketch the vectors 𝐀𝐁 and 𝐀𝐂 onto our diagram. If we remove the side 𝐵𝐶 from our sketch to clear some space, we can also add in the vectors 𝐀𝐁 and 𝐀𝐂. And now we can see we have the magnitude of vector 𝐀𝐁 since we have the length of our side is 4.75 and the magnitude of vector 𝐀𝐂 is also 4.75 and the angle between these two vectors is 60 degrees. So they should motivate us to try using the relationship between two vectors and the angle between them.
We recall if 𝜃 is the angle between vectors 𝐮 and 𝐯, then the cos of 𝜃 will be equal to the dot product between 𝐮 and 𝐯 divided by the magnitude of 𝐮 times the magnitude of 𝐯. And we’ve already found all of this information for our vectors 𝐀𝐁 and 𝐀𝐂. The angle between these two vectors is 60 degrees, and the magnitude of both of these vectors is equal to 4.75. So the only unknown in this equation is the dot product between them, which is exactly what we want to find in our question. Substituting these values into our equation, we get the cos of 60 degrees is equal to the dot product between vectors 𝐀𝐁 and 𝐀𝐂 divided by 4.75 times 4.75.
Now, we just multiply through by the denominator to find an equation for the dot product. Doing this, we get the dot product between vectors 𝐀𝐁 and 𝐀𝐂 is equal to the cos of 60 degrees multiplied by 4.75 squared. And if we evaluate this and give our answer as a decimal, we get 11.28125. But remember, the question wants us to give our answer to the nearest hundredth; that’s to two decimal places. So we need to look at the third decimal place to determine whether we need to round up or round down. Since this is less than five, we need to round down. This gives us our final answer of 11.28.
However, this isn’t the only way we could’ve answered this question. We could’ve tried to evaluate this dot product directly. And to do that, we’re going to need to write our vectors 𝐀𝐁 and 𝐀𝐂 out in full. We’re going to need to write them out component-wise. This is much more difficult to do and will involve some trigonometry. Let’s start by considering the vector 𝐀𝐂 on our diagram. We can see that vector 𝐀𝐂 has no vertical component. It only moves horizontally. In fact, we know how far it moves horizontally. It moves 4.75 units horizontally. So the vector 𝐀𝐂 is the vector with horizontal component 4.75 and zero vertical component.
We then want to find our vector 𝐀𝐁. This is more difficult because the vector 𝐀𝐁 has both horizontal and vertical components. However, we can do this by drawing the following right-angle triangle. We can then find the length of this right-angle triangle by using trigonometry. The vertical length is going to be 4.75 times the sin of 60 degrees, and the horizontal length is going to be 4.75 times the cos of 60 degrees. And because the vector 𝐀𝐁 has positive horizontal component and positive vertical component, these will give us the horizontal and vertical components of 𝐀𝐁. Therefore, we’ve shown that vector 𝐀𝐁 is the vector with horizontal component 4.75 times the cos of 60 degrees and vertical component 4.75 times the sin of 60 degrees.
And now that we found expressions for our vectors 𝐀𝐁 and 𝐀𝐂, we can directly calculate the dot product between these two vectors. Remember, when calculating the dot product between two vectors, we need to multiply the corresponding components and then add all of these products together. Multiplying the first two components of our vectors together, we get 4.75 cos of 60 degrees multiplied by 4.75. And then we need to add on the products of our second components. That’s 4.75 sin of 60 degrees multiplied by zero. And we can see the second term has a factor of zero, so it’s equal to zero, meaning we can simplify this to 4.75 squared times the cos of 60 degrees. And if we calculated this to the nearest hundredth, we would see it’s also equal to 11.28.
Therefore, we were able to show two different methods of finding the dot product between vectors 𝐀𝐁 and 𝐀𝐂, where 𝐴𝐵𝐶 is an equilateral triangle with side length 4.75. To the nearest hundredth, we got 11.28.