Question Video: Determining the Lower Bound on the Magnitude of a Sum of Vectors of Given Magnitudes Mathematics

If โ€–๐ฎโ€– = 5 and โ€–๐ฏโ€– = 2, what is the smallest that โ€–๐ฎ + ๐ฏโ€– can be?

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Video Transcript

If the magnitude of vector ๐ฎ is equal to five and the magnitude of vector ๐ฏ is equal to two, what is the smallest that the magnitude of the vector ๐ฎ plus the vector of ๐ฏ can be?

In this question, weโ€™re given the magnitude of two different vectors. The magnitude of ๐ฎ is five, and the magnitude of ๐ฏ is two. We need to use this information to determine the smallest possible value of the magnitude of the sum of vectors ๐ฎ and ๐ฏ.

Before we begin answering this question, weโ€™re going to assume that vectors ๐ฎ and ๐ฏ 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 itโ€™s possible to answer this question algebraically, itโ€™s very difficult. So, instead, weโ€™re going to do this graphically. Weโ€™ll 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. ๐ฎ plus ๐ฏ 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 ๐ฎ is five and the magnitude of vector ๐ฏ 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 ๐ฎ plus ๐ฏ. Thatโ€™s the smallest possible length of the line segment ๐ฎ plus ๐ฏ. 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, itโ€™s 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 ๐ฎ plus ๐ฏ will be bigger. And we can keep going. We can make ๐ฎ and ๐ฏ 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 ๐ฎ plus the vector ๐ฏ is just equal to the magnitude of vector ๐ฎ plus the magnitude of vector ๐ฏ. 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, letโ€™s 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 ๐ฎ plus ๐ฏ 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 ๐ฎ plus ๐ฏ 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 ๐ฎ and ๐ฏ together graphically. In particular, we know the magnitude of vector ๐ฎ is five. So the length of the directed line segment representing vector ๐ฎ is length five. And, similarly, the magnitude of vector ๐ฏ is two. And we can then calculate the magnitude of vector ๐ฎ plus ๐ฏ directly from this diagram. It adds to two to make five. So it has magnitude three. And this gives us two useful properties. If ๐ฎ and ๐ฏ 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 ๐ฎ and ๐ฏ 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 thatโ€™s the smallest possible magnitude of the sum of these two vectors.

Therefore, we were able to show if ๐ฎ has magnitude five and ๐ฏ has magnitude two, then the smallest possible magnitude of the vector ๐ฎ plus the vector ๐ฏ is five minus two, which is equal to three.

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