Which of the following parallelogram shows a valid way of obtaining the vector 𝐀 minus the vector 𝐁?
In this question, we’re given four possible diagrams to try and determine vector 𝐀 minus vector 𝐁. We need to determine which of these diagrams is correct. And there’s a few different ways we could go about answering this question. For example, we could look at each of the four given options and try to determine which one represents vector 𝐀 minus vector 𝐁 correctly. And this would definitely work; we could use this to get the correct answer.
However, we can simplify this method slightly by noticing all four of the given options are parallelograms. In particular, the sides of these parallelograms are vectors 𝐀 and 𝐁 or negative 𝐁. This means we can use the parallelogram rule to find an expression for the diagonal of each of these four parallelograms.
To do this, let’s first recall exactly what we mean by the parallelogram rule for vector addition. This tells us if we have two vectors 𝐮 and 𝐯, which we sketch as the sides of a parallelogram, initiating from the same point, then the vector along the diagonal of this parallelogram initiating from the same point will be vector 𝐮 plus vector 𝐯. We can use this to find expressions for the diagonals of all four of the given options. However, we do need to be careful about some of the intricacies of applying this result.
First, it’s very important to check that the vectors of the side of our parallelogram are initiating from the same point. Similarly, it’s very important to choose the correct diagonal. It needs to have the same initial point as vectors 𝐮 and 𝐯. Before we check each of the four options individually, it’s worth noting we can directly use this result to construct vector 𝐀 minus vector 𝐁. We could do this by choosing vector 𝐮 to be vector 𝐀 and vector 𝐯 to be vector negative 𝐁. If we do this, the parallelogram rule for vector addition tells us the diagonal of this parallelogram as a vector will be vector 𝐀 plus vector negative 𝐁. And we know that this simplifies to give us 𝐀 minus 𝐁.
And we can see this construction is exactly the same as the one given an option (D), the sides of the parallelogram of vectors 𝐀 and negative 𝐁, which have the same initial point. So the diagonal of this parallelogram starting at the same initial point as a vector will be 𝐀 plus negative 𝐁, which is 𝐀 minus 𝐁. However, for due diligence, let’s also check the other three given options to see why they might be incorrect.
Let’s start by finding an expression for the diagonal vector given in option (A). First, let’s look at the diagonal vector we need to find. Its initial point is on the bottom-left corner. However, vectors 𝐀 and 𝐁 have the terminal point on this corner, so we can’t yet directly apply the parallelogram rule for vector addition. However, we can recall if we multiply a vector by negative one, we switch the direction of the vector; however, we leave its magnitude unchanged. For example, if we switch the direction of vector 𝐀, we have vector negative 𝐀. And we can see its initial point is in the bottom-left corner. And it runs along the same side of this parallelogram.
We can apply the exact same results to vector 𝐁. We can see the other side of this parallelogram can be represented by negative 𝐁. We can now add these two vectors together to find an expression for the diagonal of this parallelogram. It would be negative 𝐀 plus negative 𝐁, which we can simplify to be negative 𝐀 minus 𝐁. And of course this is not equal to 𝐀 minus 𝐁. So option (A) is incorrect.
We can apply the same reasoning two options (B) and (C). In option (B), we can see that vector 𝐀 shares the same initial point as the diagonal vector. However, vector negative 𝐁 does not. So once again, we’ll multiply this vector by negative one to switch its direction, where we recall negative one times negative 𝐁 will just be vector 𝐁. And now we can directly use the parallelogram rule for vector addition to add these two vectors together. The diagonal of this parallelogram is vector 𝐀 plus vector 𝐁. And this is not vector 𝐀 minus vector 𝐁, so option (B) is incorrect.
Finally, let’s look at option (C). We can see the (C) is already in the direct form we can apply the parallelogram rule for vector addition. This tells us this diagonal of the parallelogram will be vector 𝐀 plus vector 𝐁 not vector 𝐀 minus vector 𝐁. So option (C) is also not correct. Therefore, we were able to show that option (D) is the correct construction for obtaining vector 𝐀 minus vector 𝐁.