Question Video: Using an Argand Diagram to Represent Operations on Complex Numbers Mathematics

Video Transcript

The blue vector represents the complex number 𝐳 one. The green vector represents the complex number 𝐳 two. What does the red vector represent?

Now, try not to worry too much that the axis labels are not what you’re used to. We’ll consider this to be much like an 𝑥𝑦-plane. We’ve got two vectors, given by 𝐳 one and 𝐳 two. Then, if we look at the red line, we see it travels from the initial point of 𝐳 one to the terminal point — that’s the end point — of 𝐳 two.

And when looking at vectors, I like to think about them a little bit like a metro or a tube map. Sometimes we want to travel from one station to another but can’t get directly there. And instead, we travel to an intermediate station, change trains, and travel on to the final destination. Our final destination is the same. We just had to go about it in a different way. In this case, the ultimate destination is from the point zero, zero to the point seven, one. Rather than going straight there though, we traveled from zero, zero to two, three along the vector 𝐳 one. And then we traveled from two, three to seven, one along the vector 𝐳 two.

In vector form, we say that our journey is 𝐳 one plus 𝐳 two. Now, we can check this by looking at the components of each vector. 𝐳 one is given by the vector two, three. To travel from the initial to the terminal point of our second vector, we travel five right and two down. So its components are five, negative two.

We said that we thought the red vector was the sum of these. Two, three plus five, negative two. Well, we find the sum of these vectors by adding their component parts. We add two and five to give us seven. Then we add three and negative two to get one. So 𝐳 one plus 𝐳 two is seven, one. And if we compare that to the red vector, we see that is also seven, one. The red vector represents 𝐳 one plus 𝐳 two.