Video Transcript
Square ๐ด๐ต๐ถ๐ท has side 10 centimeters. What is the dot product between the vector ๐๐ and ๐๐?
In this question, weโre given some information about the square ๐ด๐ต๐ถ๐ท. Weโre told the side lengths of this square are 10 centimeters. We need to use this to determine the dot product between the vectors representing two of its sides, the vector ๐๐ and the vector ๐๐.
Letโs start by sketching a picture of our square ๐ด๐ต๐ถ๐ท with side length 10 centimeters. Thereโs actually a few different ways we could evaluate this dot product. One way of doing this will be to write our vectors ๐๐ and ๐๐ component-wise from our diagram. For example, we can see the vector that goes from ๐ด to ๐ต has no horizontal component and its vertical component will be 10 centimeters. Therefore, the vector from ๐ด to ๐ต could be represented by having horizontal component zero and vertical component 10, because to go from point ๐ด to point ๐ต, we increase the vertical component by 10 centimeters.
We can do exactly the same for our vector ๐๐. To go from the point ๐ต to the point ๐ถ, our vertical component will be zero. However, we increase our horizontal component by 10. So the vector from ๐ต to ๐ถ can be represented by the vector 10, zero.
Now we can find the dot product between these two vectors directly. To do this, we need to recall how exactly we calculate the dot product between two vectors. Remember, to do this, we multiply the corresponding components together and then add the results. So we start by multiplying the first components of our vectors together. Thatโs zero multiplied by 10. And then we add to this the product of the second components of our vectors. Thatโs 10 multiplied by zero. And of course we can calculate this. Itโs equal to zero.
However, this isnโt the only way we couldโve evaluated this expression. We know a formula involving the dot product between two vectors and the angle between them. We recall if ๐ is the angle between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ must be equal to the dot product between ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ. So another way of evaluating the dot product given to us in the question is to find the magnitude of our two vectors and the angle between them. Then we can rearrange this equation and solve for the dot product.
Letโs start by finding the magnitude of our two vectors. Thatโs the vector ๐๐ and the vector ๐๐. This notation tells us the vector ๐๐ is the vector from ๐ด to ๐ต and the vector ๐๐ is the vector from ๐ต to ๐ถ. And we can see from our diagram both of these are going to be side lengths in our square. And weโre told in the question that the square has side length 10 centimeters. So we can start by saying the magnitude of ๐๐ and the magnitude of ๐๐ are going to be 10 centimeters.
Now, we need to find the angle between our two vectors. Letโs start by drawing these vectors on our diagram. Weโll start by drawing the vector ๐๐. Thatโs the vector from ๐ด to ๐ต. And then weโll also draw on the vector from ๐ต to ๐ถ.
Now, we need to be careful here. Itโs very tempting to call the angle ๐ด๐ต๐ถ the angle between these two vectors. However, this would be incorrect. To find the angle between our two vectors, our vectors must start at the same point. And we can see this is not true in our diagram. The vector ๐๐ starts at ๐ด and ends at ๐ต and the vector ๐๐ starts at ๐ต and ends at ๐ถ. So weโre going to need to move one of our vectors. Letโs move the vector ๐๐.
Remember, a vector is an object with magnitude and direction. So if the magnitude and direction of two vectors are the same, then the vectors are the same. Because ๐ด๐ต๐ถ๐ท is a square, we know the length of ๐ด๐ท is equal to 10 and we also know that the vectors ๐๐ and ๐๐ are parallel. Therefore, what weโve just shown is the vector from ๐ด to ๐ท and the vector from ๐ต to ๐ถ have the same magnitude and direction. They represent the same vector. So the angle between our vectors ๐๐ and ๐๐ is represented by angle ๐ท๐ด๐ต. And of course this is a right angle. So we know itโs equal to 90 degrees.
Now that weโve found the magnitude of our vectors and the angle between them, we can substitute these values into our formula. We get the cos of 90 degrees is equal to the dot product between our vectors ๐๐, ๐๐ divided by 10 times 10. And if we start evaluating this expression, we get something interesting. The cos of 90 degrees is equal to zero. So the right-hand side of this equation must be equal to zero. And therefore, our numerator is equal to zero.
This also proves the dot product between these two vectors is equal to zero. And it illustrates a nice use of one of our properties. We know if ๐ฎ and ๐ฏ are perpendicular vectors, then the dot product between ๐ฎ and ๐ฏ will be equal to zero. And this is because ๐ฎ and ๐ฏ being perpendicular means the angle between them is 90 degrees. And if ๐ is 90 degrees, the cos of 90 degrees is equal to zero. So the right-hand side of this equation must be equal to zero. And the only way this can be true is if the dot product between ๐ฎ and ๐ฏ is equal to zero.
Therefore, we were able to show two different ways of finding the dot product between vectors ๐๐ and ๐๐, which were side lengths of the square ๐ด๐ต๐ถ๐ท with side length 10 centimeters. In both cases, we were able to show it was equal to zero.
