The additive inverse of the vector 𝚩𝐂 is the vector blank. Option (A) the vector 𝚩𝐂, option (B) the vector negative 𝐂𝚩, option (C) the zero vector, or option (D) the vector 𝐂𝚩.
In this question, we’re asked to determine which of four vectors is the additive inverse of a vector 𝚩𝐂. The first thing we’re going to need to do is recall what we mean by the additive inverse of a vector. And to do this, let’s start by recalling what the additive inverse of a number is. Let’s start with the number 𝑥. The additive inverse of 𝑥 will be the number we add to 𝑥 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 𝑦. We know that 𝑦 would have to be equal to negative 𝑥. So the additive inverse of a real number 𝑥 is negative 𝑥.
Let’s try and construct the same property, but this time with vectors. First, we’re 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 𝐯 added to the zero vector of the same dimension will just be equal to this vector 𝐯. Therefore, to find the additive inverse of any vector 𝐯, we want to find the vector 𝐮 we add to 𝐯 to get the zero vector. In our question, we want to find the additive inverse of the vector 𝚩𝐂.
There’s a few different ways of finding the additive inverse of 𝚩𝐂. We’ll start by doing this graphically. Let’s start with two points, 𝚩 and 𝐂, and let’s assume that these two points are not the same. We can then graphically represent the vector 𝚩𝐂 by connecting 𝚩 and 𝐂 with a line segment, and we know our vector starts at 𝚩 and ends at 𝐂. We want to find the vector which when we add to this vector of 𝚩𝐂 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 𝚩𝐂.
So for 𝐮 to be the additive inverse of 𝚩𝐂, 𝚩𝐂 plus 𝐮 must be equal to zero vector. So in our diagram, the pink vector, or the vector 𝚩𝐂 plus 𝐮, must be the zero vector. It must have zero magnitude. And the only way this can happen is if our vector 𝐮 has exactly the same magnitude as vector 𝚩𝐂 but points in the opposite direction. And we can represent this as the vector 𝐂𝚩. Therefore, we’ve shown if 𝐮 is the additive inverse of the vector 𝚩𝐂, then 𝐮 should be equal to the vector 𝐂𝚩, which was our option (D).
But it’s worth pointing out this is not the only way we could’ve answered this question. If we go back to our definition of the additive inverse of two vectors, then we could subtract the vector 𝐮 from both sides of the equation. Since we know the vector 𝐮 subtracted from itself is equal to the zero vector, we get that 𝐯 is equal to negative 𝐮. We can then just apply this to the vector given to us in the question to find its additive inverse. We know if 𝐮 is the additive inverse of the vector 𝚩𝐂, then 𝚩𝐂 should be equal to negative 𝐮.
Of course, we can solve for 𝐮 by just multiplying through by negative one. We get that the additive inverse of the vector 𝚩𝐂 is the vector negative 𝚩𝐂. And it is worth pointing out all this is really saying is that 𝚩𝐂 plus negative 𝚩𝐂 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 𝚩𝐂. However, we know the answer we’re supposed to get is the vector 𝐂𝚩. 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 𝚩𝐂 is going to be the vector 𝐂𝚩.
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 𝚩𝐂. In our case, we showed this was the vector 𝐂𝚩, which was our option (D).