Question Video: Geometric Interpretation of Multiplication by 𝑖 | Nagwa Question Video: Geometric Interpretation of Multiplication by 𝑖 | Nagwa

# Question Video: Geometric Interpretation of Multiplication by π Mathematics • Third Year of Secondary School

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Four complex numbers π§β, π§β, π§β, and π§β are on the Argand diagram. 1) Find the image of the points π§β, π§β, π§β, and π§β under a transformation that maps π§ to ππ§. 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 π§ one, π§ two, π§ three, and π§ four are shown on the Argand diagram. Part 1) Find the image of the points π§ one, π§ two, π§ three, and π§ four under a transformation that maps π§ to ππ§. Part 2) By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

Weβre looking to find the transformation that maps π§ to ππ§. To do this, weβre going to first need to find the complex numbers π§ one, π§ two, π§ three, and π§ four. Remember, the horizontal axis represents the real part of a complex number. And the vertical axis represents the imaginary part. π§ one has Cartesian coordinates three, zero. So in complex number form, itβs three plus zero π, which is just three. π§ two is two plus three π. π§ three is negative two minus one π. π§ four has Cartesian coordinates zero, negative one. So as a complex number, itβs negative π.

Next, weβre going to multiply each of these numbers by π, remembering of course that π squared equals negative one. This means that ππ§ one is three π. π§ two is two π plus three π squared. And since π squared is negative one, thatβs negative three plus two π. And in the same way, ππ§ three is one minus two π. And ππ§ four is one. We now need to plot these points on the Argand diagram.

We can see that ππ§ one has Cartesian coordinates zero, three. Thatβs here. ππ§ two has Cartesian coordinates negative three, two. Thatβs here. ππ§ three is here. And ππ§ four is here. We can see that π§ one has moved a quarter of a turn here. π§ two has moved a quarter of a turn, as had π§ three and π§ four. And we can see that the transformation that maps π§ to ππ§ is a rotation about the origin in a counterclockwise direction by π by two radians.

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