Question Video: Identifying Equivalent Vectors in a Figure Mathematics • First Year of Secondary School

Video Transcript

In the given figure, 𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon with center 𝑚. Complete the following. The vector 𝐀𝐁 is equivalent to what? Is it (A) the vector 𝐌𝐄, (B) the vector 𝐅𝐌, (C) the vector 𝐁𝐌, (D) the vector 𝐃𝐂, or (E) the vector 𝐃𝐌?

We say that two vectors are equal or equivalent if they have the same magnitude and direction. And this is regardless of where they lie. For example, let vector 𝐚 be equal to one, three and 𝐛 be equal to one, three. These both represent a movement of one unit right and three units up. They’re equivalent vectors. We say vector 𝐚 is equal to vector 𝐛.

Now, we’re looking to find a vector which is equivalent to the vector 𝐀𝐁. We don’t really know officially the magnitude and direction of vector 𝐀𝐁. But we can infer its equivalent vectors from our diagram using a bit of geometrical reasoning. The vector 𝐀𝐁 moves from 𝐀 to 𝐁, as shown. And we know in a regular hexagon, opposite sides are parallel and equal in length. We also know that we can split a regular hexagon into six equilateral triangles about the center. And in doing so, we know that these sides are parallel to these sides.

Since the triangles are equilateral, this means each of these sides must also be equal in length. And actually, this means the vector 𝐀𝐁 has several equivalents. We move from left to right, and we see that it’s equivalent to the vector 𝐅𝐌. Similarly, it’s equivalent, it’s equal to the vector 𝐌𝐂. And it has one further equivalent vector. The vector 𝐄𝐃 is equal in magnitude and direction. Of those vectors in our list, we can see the correct answer is (B). It’s the vector 𝐅𝐌.

Note that had the vector say 𝐌𝐅 been listed, we could not have deduced that 𝐀𝐁 and 𝐌𝐅 are equal. We could, however, say that the vector 𝐀𝐁 is equal to the negative vector 𝐌𝐅, since changing the sign changes the direction in which we move.

Now, there are actually a bunch of other equivalent vectors in our regular hexagon. Take the vector 𝐀𝐅 for example. It’s equal in direction and magnitude to the vector 𝐁𝐌 for the same reasons. It’s equivalent to the vector 𝐌𝐄. And each of these is also equivalent to the vector 𝐂𝐃.