yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


2m read
·Nov 10, 2024

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.

More Articles

View All
Finding Humanity Through Photos | National Geographic
[Music] Creativity and rhythm, I think, go hand in hand for me. Once I get a rhythm, then breaking that is where I get inspiration. [Applause] As a little kid, I was always catching critters and snakes. Once I got a camera, that grew into photographing th…
What is Breakthrough Starshot?
The closest star system to our own Sun is Alpha Centauri, and nearly 4.5 light-years away from the Sun, they consist of three stars: Alpha Centauri A and B, who happen to form a binary star system as they orbit around each other in a cosmic dance. In Alph…
Valley of the Boom: Trailer #1 | National Geographic
That little A and At? See, that’s what I said. Mm-hm. Um, Katie said she thought it was “about.” Yeah. Oh. MAN: But I’d never heard it. KATIE COURIC: Or around or about. MAN: I’d never heard it said. I’d always seen the mark but never heard it said. Y…
Announcing O'Leary Fine Wines
[Music] And we are back now with the new edition of Shar Tank. Your life, we have two entrepreneurs ready to go head-to-head. Kevin Oer from Shark Tank is here. We’ve already seen him double Dutch; it’s one of his many, many talents. Also wearing that Smi…
How to Make the Maximum Amount of Money Possible
So if you want to make the maximum amount of money possible, like, if you just, if you just want to get rich over your life, and you want to do it in a deterministic, predictable way, what you would do is you would basically stay on the bleeding edge of t…
Homeroom with Sal & Dan Tieu & Sophie Turnbull - Wednesday, August 19
Hi everyone, Sal here from Khan Academy. Welcome, uh, again for joining our homeroom live stream. Uh, before we get into what’s going to be a really fun conversation with some internal Khan Academy team members to talk about all the new things we have for…