Question Video: Arithmetic Operations of Vectors Mathematics

Video Transcript

We’ve got 𝐴𝐵; it’s the vector three 𝑖 plus four 𝑗. 𝑋𝑌 is the vector negative two 𝑖 plus three 𝑗. And we’ve just got to add those two vectors together.

So the first stage is just to take vector 𝐴𝐵 and add vector 𝑋𝑌. And all we have to do is add the 𝑖s together first, and then add the 𝑗s together second. So three 𝑖 add negative two 𝑖 is just one 𝑖, so that’s 𝑖. And four 𝑗 add positive three 𝑗 is seven 𝑗. So there’s our answer, 𝑖 plus seven 𝑗. Simple as that, add the 𝑖-components together, add the 𝑗-components together, look out for the negative signs and- when you’re doing those calculations; but otherwise, that’s a pretty straightforward process.

So let’s just visualise that example. So we had three 𝑖 plus four 𝑗. So we’ll be going positive three 𝑖 and then positive four 𝑗, this is the 𝑎𝑏 vector. So there we are. We’ve just laid that down at the origin. We could’ve laid it anywhere on the-on the graph. Now adding vectors, we just lay them end-to-end. So what we were adding, 𝑥𝑦, was negative two 𝑖 plus three 𝑗. So we’re effectually starting 𝑥 from 𝐵, so 𝑥 lays on top of 𝐵 and we’re going negative two 𝑖. So we’re going two in the negative 𝑥-direction and we’re going positive three 𝑗, up to here.

So adding vectors is just a matter of laying them end-to-end on the graph. So we laid 𝐴𝐵 down, which started here and ended here, and then we just added 𝑋𝑌 to the end of that, laid that onto the end. So that started from where we just finished off and then ended up here. So the resultant vector is this green one here. And to get from the beginning of the green vector to the end of the green vector, we had to go positive one in the 𝑥-direction. So that’s one 𝑖, or just 𝑖. And in the 𝑦-direction, we’re going up seven all the way up here. So that’s plus seven 𝑗.

So when you’re doing these questions, it really is just a matter of adding the 𝑥-components together, adding the 𝑦-components together, and coming up with a simple answer. You don’t need to do all this graphical checking. But I’m just hoping that that’s giving you some extra insight into the process and why it works.