Video: Geometric Interpretation of Multiplication by 𝑖

Four complex numbers 𝑍₁, 𝑍₂, 𝑍₃, and 𝑍₄ are shown on the Argand diagram. 1) Find the image of the points 𝑍₁, 𝑍₂, 𝑍₃, and 𝑍₄ under a transformation that maps 𝑍 to 𝑖𝑍. 2) By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

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Video Transcript

Four complex numbers 𝑍 one, 𝑍 two, 𝑍 three, and 𝑍 four are shown on the Argand diagram. Part 1) Find the image of the points 𝑍 one, 𝑍 two, 𝑍 three, and 𝑍 four under a transformation that maps 𝑍 to 𝑖𝑍. Part 2) By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

We’re looking to find the transformation that maps 𝑍 to 𝑖𝑍. To do this, we’re going to first need to find the complex numbers 𝑍 one, 𝑍 two, 𝑍 three, and 𝑍 four. Remember, the horizontal axis represents the real part of a complex number. And the vertical axis represents the imaginary part. 𝑍 one has Cartesian coordinates three, zero. So in complex number form, it’s three plus zero 𝑖, which is just three. 𝑍 two is two plus three 𝑖. 𝑍 three is negative two minus one 𝑖. 𝑍 four has Cartesian coordinates zero, negative one. So as a complex number, it’s negative 𝑖.

Next, we’re going to multiply each of these numbers by 𝑖, remembering of course that 𝑖 squared equals negative one. This means that 𝑖𝑍 one is three 𝑖. 𝑍 two is two 𝑖 plus three 𝑖 squared. And since 𝑖 squared is negative one, that’s negative three plus two 𝑖. And in the same way, 𝑖𝑍 three is one minus two 𝑖. And 𝑖𝑍 four is one. We now need to plot these points on the Argand diagram.

We can see that 𝑖𝑍 one has Cartesian coordinates zero, three. That’s here. 𝑖𝑍 two has Cartesian coordinates negative three, two. That’s here. 𝑖𝑍 three is here. And 𝑖𝑍 four is here. We can see that 𝑍 one has moved a quarter of a turn here. 𝑍 two has moved a quarter of a turn, as had 𝑍 three and 𝑍 four. And we can see that the transformation that maps 𝑍 to 𝑖𝑍 is a rotation about the origin in a counterclockwise direction by πœ‹ by two radians.

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