Video Transcript
Select all the statements that must be true if the vector 𝐮 and the vector 𝐯 are equivalent vectors. Option (A) vector 𝐮 and vector 𝐯 have the same initial point. Option (B) vector 𝐮 and vector 𝐯 have the same terminal point. Option (C) the magnitude of vector 𝐮 is equal to the magnitude of vector 𝐯. Option (D) the initial point of vector 𝐮 is the terminal point of vector 𝐯. And option (E) the initial point of vector 𝐯 is the terminal point of vector 𝐮.
In this question, we’re given two vectors 𝐮 and 𝐯 and we’re told that these are equivalent vectors. And we’re given five possible statements about these two vectors. We need to select all of the statements which have to be true.
To do this, let’s first recall exactly what it means for two vectors to be equivalent. And in fact, there’s two different ways of defining equivalent vectors. And both of these are useful in different situations. And in fact, you’ll be able to use either of these to answer this question. The first way we could say two vectors are equivalent is if they have the same magnitude and direction. Any two vectors which are equivalent have to have the same magnitude and direction. And any two vectors with the same magnitude and direction must be equivalent.
However, there’s a second way of defining two equivalent vectors, which is useful in different situations. We know if all of the corresponding components of two vectors are equal and these two vectors have the same dimension, then these two vectors are equivalent. Similarly, if two vectors are equivalent, then all of the corresponding components will be equal and they’ll be of the same dimension. As a rule of thumb, usually the first definition is more useful when we’re thinking graphically and the second definition is more useful when we’re given the vectors in terms of components. However, we can use both of them in both situations if we prefer.
Using this, we can immediately notice something interesting about option (C). Option (C) says that the magnitude of vector 𝐮 needs to be equal to the magnitude of vector 𝐯. And that’s exactly in part of the definition. For two vectors to be equal, their magnitude must be equal and their direction must be equal. So option (C) has to be true. Because vector 𝐮 is equal to vector 𝐯, their magnitudes must be equal.
But that’s not all because the question wants us to select all of the statements that have to be true. In fact, we’ll see that all four of the remaining statements are not necessarily true. And it will be easiest to show this graphically. Let’s start with a pair of axis 𝑥 and 𝑦. And we’ll start with the unit directional vector in the 𝑥-direction starting at the origin. We’ll mark this 𝐢.
The initial point of vector 𝐢 in our diagram is the origin, and the terminal point will have coordinates one, zero. But now we can ask an interesting question. What if we have the exact same vector, however, this time its initial point is the point zero, one? The magnitude of both of these vectors is the same. They’re both unit vectors; they both have length one. And we can see in our diagram the direction of both of these vectors is the same. We know they both point in the positive horizontal direction and not at all in the vertical direction.
So both of these vectors represent the same vector. Both of these are the vector 𝐢. And we’ll mark a few points just onto our diagram. We’ll mark the origin. And we’ll also mark one onto our 𝑥-axis. Now we can specifically see the initial point and the terminal point of both of these two vectors. Firstly, we can see that these two vectors don’t have the same initial point. But we do know that they’re equal, so option (A) can’t be true.
Next, we can also see they don’t have the same terminal point. But we do know these two vectors are equal, so option (B) can’t be true either. Similarly, we can see that the initial point of one vector is not the terminal point of another vector. So both option (D) and option (E) are also not true. It wouldn’t matter which one we called vector 𝐮 and which one we called vector 𝐯.
And this brings us to a very interesting point about vectors. It can be very useful to think about vectors in terms of their initial point and terminal point because this tells us their magnitude and their direction. However, if we only know the magnitude and direction of a vector, we don’t know its initial point or its terminal point. And it’s this exact reason that only option (C) is going to be true if 𝐮 and 𝐯 are equivalent vectors.
Therefore, we were able to show if 𝐮 and 𝐯 are two equivalent vectors, then of all the options shown, only option (C), the magnitude of 𝐮 being equal to the magnitude of 𝐯, must be true.
