### 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 ππ. Part 2) By plotting these
points on an Argand diagram, or otherwise, give a geometric interpretation of
the transformation.

Weβre looking to find the
transformation that maps π to ππ. To do this, weβre 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, itβs
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, itβs
negative π.

Next, weβre going to multiply
each of these numbers by π, remembering of course that π squared equals
negative one. This means that ππ one is
three π. π two is two π plus three π
squared. And since π squared is
negative one, thatβs negative three plus two π. And in the same way, ππ three
is one minus two π. And ππ four is one. We now need to plot these
points on the Argand diagram.

We can see that ππ one has
Cartesian coordinates zero, three. Thatβs here. ππ two has Cartesian
coordinates negative three, two. Thatβs here. ππ three is here. And ππ 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 ππ is a rotation about the origin in a
counterclockwise direction by π by two radians.