Some vectors are drawn to scale on a square grid. Which color vector shows the resultant of the black vectors 𝐴 and 𝐵?
Okay, so in this question, we’ve been given a square grid. That is a grid where the spaces between lines in this direction is the same as the spaces between lines in this direction. And as well as this, drawn on the square grid, we have two black vectors 𝐴 and 𝐵. And we want to find the resultant of these two black vectors. So, how do we go about finding the resultant of two vectors? Well, we can do this by adding the two vectors together. But then, adding two vectors together is not as simple as adding two numbers together.
So, how do we add vectors together? Well, we do this by using what is known as the tip-to-tail method. Essentially, we slide one of the vectors along until its tail meets the tip of the other vector. So, for example, we want to add this vector and this vector. What we do is we take one of the vectors. So let’s say we take this one. And we slide it along until its tail meets the tip of the other vector. In other words, we slide this whole vector until its tail is meeting the tip of the first vector. And that looks something like this. Now, note that the direction in which the second vector is pointing has not changed and neither has the magnitude or length of the vector. All that’s changed is the position of the vector. Initially, we’d drawn it here. But we’ve simply moved it along to here.
So anyway, now that we’ve joined up the tip of the first vector to the tail of the second vector, we can calculate the resultant of these two vectors or, in other words, the result of adding these two vectors. The way to do this is to take the starting point of our resultant vector as the tail of the first vector and the ending point of our resultant vector as the tip of the second vector. Then, we simply draw in a vector starting here and ending here. And this pink vector we’ve drawn in is the resultant of these two orange vectors. And we’re going to apply the same logic to our two black vectors here. We need to slide one of the black vectors, either vector 𝐴 or vector 𝐵, until its tail meets the tip of the other vector.
So, let’s imagine we’re going to slide vector 𝐵. Now, vector 𝐵 actually begins here. This is its tail currently and its tip is here. On our grid, that’s equivalent to moving one, two, three spaces to the left and one, two, three spaces down. Therefore, if we want to draw vector 𝐵, but this time starting at the tip of vector 𝐴, then we need to ensure that our new drawing of vector 𝐵 moves along one, two, three spaces to the left and one, two, three spaces down. In other words, our newly drawn vector 𝐵 now begins at this point, which was originally the tip of vector 𝐴 and ends at this point here.
Now, all that’s left for us to do is to realise that our resultant vector is going to start at the tail of vector 𝐴, which is the first vector we’re adding, and finish at the tip of vector 𝐵, which is the second vector we’re adding. In other words, that’s this vector here. That vector happens to be the green vector in the original diagram.
And hence, as the answer to our question, we can say that the green color vector shows the resultant of the black vectors 𝐴 and 𝐵.