Video Transcript
The additive inverse of the vector ๐ฉ๐ is the vector blank. Option (A) the vector ๐ฉ๐, option (B) the vector negative ๐๐ฉ, option (C) the zero vector, or option (D) the vector ๐๐ฉ.
In this question, weโre asked to determine which of four vectors is the additive inverse of a vector ๐ฉ๐. The first thing weโre going to need to do is recall what we mean by the additive inverse of a vector. And to do this, letโs start by recalling what the additive inverse of a number is. Letโs start with the number ๐ฅ. The additive inverse of ๐ฅ will be the number we add to ๐ฅ to give us zero. This is because zero is what we call the additive identity. If we add it to any number, we just end up with that same number, and we can solve this for ๐ฆ. We know that ๐ฆ would have to be equal to negative ๐ฅ. So the additive inverse of a real number ๐ฅ is negative ๐ฅ.
Letโs try and construct the same property, but this time with vectors. First, weโre going to need to think what do we mean by the additive identity of a vector. Either by thinking graphically or in terms of components, we know that this will be the zero vector. We know for any vector ๐ฏ added to the zero vector of the same dimension will just be equal to this vector ๐ฏ. Therefore, to find the additive inverse of any vector ๐ฏ, we want to find the vector ๐ฎ we add to ๐ฏ to get the zero vector. In our question, we want to find the additive inverse of the vector ๐ฉ๐.
Thereโs a few different ways of finding the additive inverse of ๐ฉ๐. Weโll start by doing this graphically. Letโs start with two points, ๐ฉ and ๐, and letโs assume that these two points are not the same. We can then graphically represent the vector ๐ฉ๐ by connecting ๐ฉ and ๐ with a line segment, and we know our vector starts at ๐ฉ and ends at ๐. We want to find the vector which when we add to this vector of ๐ฉ๐ we get the zero vector. To do this, we need to recall how we add two vectors together graphically. We need the terminal point of our first vector to be equal to the initial point of our second vector. Then, graphically, traveling along both of these vectors will be the same as adding the two vectors together. We want to use this to find the additive inverse of our vector ๐ฉ๐.
So for ๐ฎ to be the additive inverse of ๐ฉ๐, ๐ฉ๐ plus ๐ฎ must be equal to zero vector. So in our diagram, the pink vector, or the vector ๐ฉ๐ plus ๐ฎ, must be the zero vector. It must have zero magnitude. And the only way this can happen is if our vector ๐ฎ has exactly the same magnitude as vector ๐ฉ๐ but points in the opposite direction. And we can represent this as the vector ๐๐ฉ. Therefore, weโve shown if ๐ฎ is the additive inverse of the vector ๐ฉ๐, then ๐ฎ should be equal to the vector ๐๐ฉ, which was our option (D).
But itโs worth pointing out this is not the only way we couldโve answered this question. If we go back to our definition of the additive inverse of two vectors, then we could subtract the vector ๐ฎ from both sides of the equation. Since we know the vector ๐ฎ subtracted from itself is equal to the zero vector, we get that ๐ฏ is equal to negative ๐ฎ. We can then just apply this to the vector given to us in the question to find its additive inverse. We know if ๐ฎ is the additive inverse of the vector ๐ฉ๐, then ๐ฉ๐ should be equal to negative ๐ฎ.
Of course, we can solve for ๐ฎ by just multiplying through by negative one. We get that the additive inverse of the vector ๐ฉ๐ is the vector negative ๐ฉ๐. And it is worth pointing out all this is really saying is that ๐ฉ๐ plus negative ๐ฉ๐ is the zero vector. And this is the exact statement you would get if you tried to think about this component-wise.
But of course, this is not quite enough to answer our question because in this case we get the answer negative ๐ฉ๐. However, we know the answer weโre supposed to get is the vector ๐๐ฉ. Instead, we need to use one more piece of information we know about vectors. When you multiply a vector by negative one, you switch its directions. So negative ๐ฉ๐ is going to be the vector ๐๐ฉ.
Therefore, in this question, we were able to determine exactly what is meant by the additive inverse of a vector. And we were able to use this to find multiple different methods of finding an expression for the additive inverse of the vector ๐ฉ๐. In our case, we showed this was the vector ๐๐ฉ, which was our option (D).
