Multiplying complex numbers graphically example: -3i | Precalculus | Khan Academy

Suppose we multiply a complex number z by negative 3i, and they show us z right over here. Plot the point that represents the product of z and negative 3i. So pause this video and see if you can work through that.

All right, now let's do it step by step. First, I want to think about where 3z would be. Well, 3z would have the same angle as z, but its absolute value, or its modulus, would be three times larger. So you'd be going in this direction, but it'd be three times further. So that's one times its modulus, that's two times its modulus, that's three times its modulus, or it's three times its absolute value. So 3z would be right over here.

Now, what about negative 3z? Well, if you multiply it by a negative, it's just going to flip it around. You could think about it as flipping it at 180 degrees, but it's going to have the same modulus. So instead of being right over here at 3 in this direction, it's going to be 1, 2, 3 in this direction, right over here. So that is negative 3z.

Now, perhaps most interestingly, what happens when you multiply by i? So if we have negative 3i times z, now which is exactly what they want us to figure out, well let's think about what happens if you had 1. If you multiplied it by i, so 1 times i becomes 1i, so it goes over there. What if you then took 1i and multiplied it by i? Well then you have negative 1. What if you took negative 1 and you multiplied it by i? Well then now you have negative 1i.

So notice every time we multiply by i, we are rotating by 90 degrees. So over here, if we take negative 3z and multiply it by i, you're just going to rotate 90 degrees, and you're going to get right over there. So this is negative 3i times z, which is exactly what we were looking for.