Video: Geometric Interpretation of Multiplication by a Real Number

Three complex numbers 𝑧₁, 𝑧₂, and 𝑧₃ are represented on the Argand diagram. Find the image of points 𝑧₁, 𝑧₂, and 𝑧₃ under the transformation that maps 𝑧 to 2𝑧. By plotting these points on an Argand diagram,or otherwise, give a geometric interpretation of the transformation.

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Video Transcript

Three complex numbers 𝑧 one, 𝑧 two, and 𝑧 three are represented on the Argand diagram. Find the image of points 𝑧 one, 𝑧 two, and 𝑧 three under the transformation that maps 𝑧 to two 𝑧. By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

We begin by recalling the general form of a complex number. A complex number 𝑧 is of the form π‘Ž plus 𝑏𝑖. π‘Ž and 𝑏 are real numbers. And we say that π‘Ž is the real part of our complex number, whilst 𝑏 is the imaginary part. So we’ll begin by writing down the three complex numbers we have.

Firstly, we have 𝑧 one. It lies on the positive real axis. In fact, it can be represented by the point whose Cartesian coordinates are two, zero. And so this means its real part is two and its imaginary part is zero. And so the complex number 𝑧 one is just two. Then we have 𝑧 two. And that lies on the negative imaginary axis. It’s represented by the point with Cartesian coordinates zero, negative one. So its real part is zero and its imaginary part, the coefficient of 𝑖, must be negative one. And we can, therefore, say that 𝑧 two is just negative 𝑖.

Then we have the complex number 𝑧 three, which sits in the second quadrant over here. This time it can be represented by the point whose Cartesian coordinates are negative three, two. And so the real part must be negative three and the imaginary part must be two. 𝑧 three is negative three plus two 𝑖.

So the first part of this question tells us to find the image of the points that represent our complex numbers under a transformation that maps 𝑧 to two 𝑧. So we go back to the general form of our complex number. 𝑧 is π‘Ž plus 𝑏𝑖. So what is two 𝑧? Well, it follows that it is, in fact, equal to two times π‘Ž plus 𝑏𝑖. And so if we distribute the two over our parentheses, we see that two 𝑧 is equal to two π‘Ž plus two 𝑏𝑖. It follows then that to map 𝑧 onto two 𝑧, we simply take the real part and we double it and then we take the imaginary part and we also multiply that by two.

So we take the image of 𝑧 one. And we take the value two, and we multiply it by two. Two times two is equal to four. And so the image of 𝑧 one, which is 𝑧 one prime, must be equal to four. Then we take the image of 𝑧 two. Now, in fact, the coefficient of 𝑖 here is negative one. And we said that we need to multiply this by two to find the image of 𝑧 two. So the image of 𝑧 two is two times negative one 𝑖, which is negative two 𝑖. Finally, we move on to the image of 𝑧 three.

And we’re going to begin by multiplying the real part, that’s negative three, by two. Two times negative three is negative six. And so the real part of our complex number 𝑧 three prime is negative six. Then we multiply the imaginary part of this complex number by two. So that’s two times two, which is four. The imaginary part of the image of 𝑧 three is four. And so we see the image of 𝑧 three, which is 𝑧 three prime, must be equal to negative six plus four 𝑖.

The second part of this question asks us to give a geometric interpretation of that transformation. And so we’re going to plot the image of each of our points on our Argand diagram. Let’s begin with 𝑧 one prime, the image of 𝑧 one. This has a real part of four and an imaginary part of zero. So it’s represented by the point whose Cartesian coordinates are four, zero. Then we look at 𝑧 two prime. This has a real part of zero and an imaginary part of negative two. So it lies here, at the point whose Cartesian coordinates are zero, negative two. And then finally, we have 𝑧 three prime. That has a real part of negative six and an imaginary part of four. So it’s all the way over here. It has Cartesian coordinates negative six, four.

So what is a geometric interpretation of the transformation? Well, if we look really carefully, we see that each point appears to be double the distance away from the origin, from the point zero, zero. And of course, this makes a lot of sense. If we double both the real and imaginary parts of our complex number, we would expect it to be double the distance away from the point zero, zero. And this means we must have a dilation or an enlargement, depending on where you live in the world. Now, of course, when we describe a dilation, we also give a center for the dilation or enlargement and a scale factor. So here the transformation represents a dilation with scale factor two centered at the origin.

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