Question Video: Properties of Vectors in the Shape of a Parallelogram Mathematics

Video Transcript

Fill in the blank: In the parallelogram 𝐴𝐵𝐶𝐷, the vector 𝐀𝐁 plus the vector 𝐀𝐃 is equal to what.

We’re looking at some geometric properties of vectors. So in order to understand what’s going on here, let’s begin by sketching a parallelogram 𝐴𝐵𝐶𝐷. Here is our parallelogram. We’re ready to label the vertices. Let’s define this vertex, the bottom-left corner, to be vertex 𝐴. Then, since the parallelogram is said to be 𝐴𝐵𝐶𝐷, one of the vertices adjacent to 𝐴 must be vertex 𝐵. Let’s choose this one. Then, the remaining vertex adjacent to 𝐵 is vertex 𝐶, meaning the final vertex is vertex 𝐷. It’s worth noting of course that we could have moved in the opposite direction. It’s just important that we follow the order the letters are given: 𝐴, 𝐵, 𝐶, then 𝐷.

Let’s now identify the relevant vectors. The first vector is vector 𝐀𝐁. This is the vector that describes the movement that takes us from point 𝐴 to point 𝐵 along a straight line. And so we can denote this using the yellow arrow. This is vector 𝐀𝐁. Next, we’re interested in the vector 𝐀𝐃. This is the vector that takes us directly from point 𝐴 to point 𝐷 along the straight line. So we denote this vector using the pink arrow as shown. Then, if we’re finding the sum of two vectors, we call that their resultant. And if we think about each vector as being like a journey, the resultant is essentially the final journey that we end up taking.

So, in this case, we move from point 𝐴 to point 𝐵 and then from point 𝐴 to point 𝐷. But in terms of journeys, this doesn’t make a lot of sense. So we can use some of the properties of parallelograms to help us understand what’s happening. We know that opposite sides in a parallelogram are both equal in length and parallel. And of course we know that vectors are defined by their magnitude and direction, not their position. This means that if we can describe the journey from 𝐴 to 𝐷 using the vector 𝐀𝐃, then we can describe the journey from 𝐵 to 𝐶 using that same vector.

And so now we can think about our resultant as taking us from point 𝐴 to point 𝐵 and then point 𝐵 to point 𝐶. So the full journey found by adding 𝐀𝐁 to 𝐀𝐃 actually takes us from point 𝐴 to point 𝐶. We can therefore describe the resultant of 𝐀𝐁 and 𝐀𝐃 using a single vector 𝐀𝐂. So, in the parallelogram 𝐴𝐵𝐶𝐷, 𝐀𝐁 plus 𝐀𝐃 is equal to the vector 𝐀𝐂.