WEBVTT
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Four complex numbers π§ one, π§ two, π§ three, and π§ four are shown on the Argand diagram.
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Part 1) Find the image of the points π§ one, π§ two, π§ three, and π§ four under a transformation that maps π§ to ππ§.
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Part 2) By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.
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Weβre looking to find the transformation that maps π§ to ππ§.
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To do this, weβre going to first need to find the complex numbers π§ one, π§ two, π§ three, and π§ four.
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Remember, the horizontal axis represents the real part of a complex number.
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And the vertical axis represents the imaginary part.
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π§ one has Cartesian coordinates three, zero.
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So in complex number form, itβs three plus zero π, which is just three.
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π§ two is two plus three π.
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π§ three is negative two minus one π.
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π§ four has Cartesian coordinates zero, negative one.
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So as a complex number, itβs negative π.
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Next, weβre going to multiply each of these numbers by π, remembering of course that π squared equals negative one.
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This means that ππ§ one is three π.
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π§ two is two π plus three π squared.
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And since π squared is negative one, thatβs negative three plus two π.
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And in the same way, ππ§ three is one minus two π.
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And ππ§ four is one.
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We now need to plot these points on the Argand diagram.
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We can see that ππ§ one has Cartesian coordinates zero, three.
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Thatβs here.
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ππ§ two has Cartesian coordinates negative three, two.
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Thatβs here.
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ππ§ three is here.
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And ππ§ four is here.
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We can see that π§ one has moved a quarter of a turn here.
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π§ two has moved a quarter of a turn, as had π§ three and π§ four.
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And we can see that the transformation that maps π§ to ππ§ is a rotation about the origin in a counterclockwise direction by π by two radians.