Video Transcript
Which of the following is equivalent to the vector from 𝐴 to 𝐵 minus the vector from 𝐷 to 𝐵? Is it option (A) the vector from 𝐴 to 𝐶? Option (B) the vector from 𝐵 to 𝐴. Option (C) the vector from 𝐵 to 𝐶. Option (D) the vector from 𝐷 to 𝐶. Or is it option (E) the vector from 𝐷 to 𝐴?
In this question, we’re given a diagram including parallelogram 𝐴𝐵𝐶𝐷. We need to use this diagram to determine which of five given vectors is equivalent to the vector from 𝐴 to 𝐵 minus the vector from 𝐷 to 𝐵. And to answer this question, let’s start by sketching the two vectors we’re given on our diagram. We sketch the vector from 𝐴 to 𝐵 to have initial point of vertex 𝐴 and terminal point of vertex 𝐵 and the vector from 𝐷 to 𝐵 to have initial point of vertex 𝐷 and terminal point of vertex 𝐵. We want to find a vector equivalent to the difference between these two vectors.
And there’re several different ways of doing this. For example, we could use a construction which tells us how to find the difference between two vectors. However, we can also just recall when we multiply a vector by negative one, we switch its direction; however, we leave its magnitude unchanged. This means we can find the vector negative one multiplied by the vector from 𝐷 to 𝐵 by switching the direction of this vector. In other words, negative one times the vector from 𝐷 to 𝐵 is equal to the vector from 𝐵 to 𝐷.
Therefore, we’ve shown the vector from 𝐴 to 𝐵 minus the vector from 𝐷 to 𝐵 is equal to the vector from 𝐴 to 𝐵 added to the vector from 𝐵 to 𝐷. And we can show these vectors on our diagram. And now, we can see the terminal point of the vector from 𝐴 to 𝐵 is coincident with the initial point of the vector from 𝐵 to 𝐷. This means we can add these two vectors together by using the triangle rule for vector addition. We can remember this rule in two different ways.
In terms of vertices, this tells us the vector from 𝑃 to 𝑄 added to the vector from 𝑄 to 𝑅 is equal to the vector from 𝑃 to 𝑅. In other words, if the terminal point of the first vector is coincident with the initial point of the second vector, then the sum of these two vectors has initial point coincident with the initial point of the first vector and terminal point coincident with the terminal point of the second vector. The vector from 𝐴 to 𝐵 added to the vector from 𝐵 to 𝐷 is the vector from 𝐴 to 𝐷. And we can add this vector onto our diagram. However, it’s worth noting we don’t need to apply the triangle rule directly to the vertices. Instead, we can remember or apply this rule by using the diagram.
We can see in the diagram we’ve drawn the vector from 𝐴 to 𝐵 and the vector from 𝐵 to 𝐷 tip to tail. This means we can combine the displacements of the two vectors, or in other words add the two vectors together, by following the arrows. Starting at point 𝐴 and applying the vector from 𝐴 to 𝐵, we will end up at the point 𝐵. But then if we apply the vector from 𝐵 to 𝐷, we will end up at the point 𝐷. So, the vector from 𝐴 to 𝐵 added to the vector from 𝐵 to 𝐷 will just be the vector from 𝐴 to 𝐷.
However, we can notice this is not one of the five given options. So, we need to find an equivalent vector to the vector from 𝐴 to 𝐷. And remember for two vectors to be equal, they need to have the same magnitude and direction. We can find a vector with equal magnitude and direction as the vector from 𝐴 to 𝐷 by using the fact that quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram.
Remember in a parallelogram opposite sides are parallel and have the same length. So, the side 𝐴𝐷 has the same length as the side 𝐵𝐶. And these two sides are parallel. This means that the vector from 𝐴 to 𝐷 and the vector from 𝐵 to 𝐶 will have the same magnitude and direction. So, we can also say the vector from 𝐴 to 𝐵 minus the vector from 𝐷 to 𝐵 is equal to the vector from 𝐵 to 𝐶, which we can see is option (C).
