Video Transcript
If the magnitude of vector ๐ฎ is equal to five and the magnitude of vector ๐ฏ is equal to two, what is the smallest that the magnitude of the vector ๐ฎ plus the vector of ๐ฏ can be?
In this question, weโre given the magnitude of two different vectors. The magnitude of ๐ฎ is five, and the magnitude of ๐ฏ is two. We need to use this information to determine the smallest possible value of the magnitude of the sum of vectors ๐ฎ and ๐ฏ.
Before we begin answering this question, weโre going to assume that vectors ๐ฎ and ๐ฏ are two-dimensional. In fact, this is not a necessary assumption. The results will be true for any two vectors of the same dimension. However, this assumption will make all of the diagrams we will use easier. Although itโs possible to answer this question algebraically, itโs very difficult. So, instead, weโre going to do this graphically. Weโll do this by recalling two things. First, to add two vectors together graphically, we need to draw them tip to tail. Then, we can add the two vectors together by using the tail of the first vector and the tip of the second vector. In other words, we follow the vectors. ๐ฎ plus ๐ฏ is given in the diagram below.
Next, we can recall the magnitude of a vector represented graphically is the length of the line segment. Therefore, since the magnitude of vector ๐ฎ is five and the magnitude of vector ๐ฏ is two, we can conclude the blue side of this triangle has length five and the orange side of this triangle has length two. And the question wants us to determine the smallest value for the magnitude of ๐ฎ plus ๐ฏ. Thatโs the smallest possible length of the line segment ๐ฎ plus ๐ฏ. In our diagram, this is in pink. We can do this by noticing the length of a side in the triangle is related to the measure of the angle opposite the side.
In fact, if we know the law of cosines, we can find a direct relationship between all four of these values. However, itโs not necessary to answer this question. Instead, we notice the bigger this angle, the bigger the length of the side will be. And in turn this means the magnitude of the vector ๐ฎ plus ๐ฏ will be bigger. And we can keep going. We can make ๐ฎ and ๐ฏ point in exactly the same direction.
Now, when we use the tip-to-tail method to add the two vectors together, we can see that the magnitude of the vector ๐ฎ plus the vector ๐ฏ is just equal to the magnitude of vector ๐ฎ plus the magnitude of vector ๐ฏ. And this is the biggest possible magnitude the sum of these two vectors can have. And this occurs when the vectors point in the same direction.
Now, letโs consider the smallest we can make this value. In exactly the same way, if we make the angle in this triangle smaller, then the side length of ๐ฎ plus ๐ฏ will be smaller. And we can keep following this process to make this side length smaller. In other words, this will make the magnitude of ๐ฎ plus ๐ฏ smaller and smaller. And in exactly the same way we showed the biggest possible magnitude of the sum of these two vectors occur when they point in the same direction, we can show that the smallest possible magnitude of the sum of these two vectors occur when they point in opposite directions.
By using the tip-to-tail method, we can add the vectors ๐ฎ and ๐ฏ together graphically. In particular, we know the magnitude of vector ๐ฎ is five. So the length of the directed line segment representing vector ๐ฎ is length five. And, similarly, the magnitude of vector ๐ฏ is two. And we can then calculate the magnitude of vector ๐ฎ plus ๐ฏ directly from this diagram. It adds to two to make five. So it has magnitude three. And this gives us two useful properties. If ๐ฎ and ๐ฏ point in the same direction, then the magnitude of the sum of these two vectors is the sum of their magnitudes. And this is the largest possible magnitude of the sum of these two vectors. And if ๐ฎ and ๐ฏ point in exactly opposite directions, then the magnitude of the sum of these two vectors is the absolute value in the difference of their magnitudes. And in this case thatโs the smallest possible magnitude of the sum of these two vectors.
Therefore, we were able to show if ๐ฎ has magnitude five and ๐ฏ has magnitude two, then the smallest possible magnitude of the vector ๐ฎ plus the vector ๐ฏ is five minus two, which is equal to three.
