SANDBOX
SANDBOX
Let's review unit vector notation quickly. Here we see the vector RA. It is shown at an angle θ with respect to the X axis. Also shown are the unit vectors î and ĵ.
You will recall that these are each of unit length and point along the X and Y axes respectively and are used to establish directions rather than positions. Our vector RA can be expressed in polar form as: |RA|@ ∠ θ. In Cartesian form that same vector would be described as: Rcosθî, Rsinθĵ where in this case we are using unit vector notation.
Let's now replace our Cartesian coordinate system with what's called the complex plane.
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What was the X axis is now the Real axis and the former Y axis is now the Imaginary axis. We can draw the same vector RA in the complex plane. If we want to think in terms of the Cartesian form, the location of point A would be given by: Rcosθ + jRsinθ where j=√-1 is the unit vector pointing along the positive imaginary axis and R = |RA|. The unit vector pointing along the real axis is, of course, the value 1.
Note: we will use j instead of i to represent imaginary numbers. We do this so that later on we won't have conflicts with the use of i to represent an electrical current.
We can also represent our vector RA in the complex plane in polar form: Rejθ. This is, of course, making use of Euler's identity: ejθ = cosθ + jsinθ.
Before moving on, let's consider some rotations of vectors in the complex plane.
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We began by drawing vector RA directly along the real axis (recall that R=|RA). Let's now rotate it 90° to create vector RB. We can see that: RB=jR. If we rotate a further 90° we get vector RC=j2R=-R. Finally, a further 90° rotation gives us vector: RD = j3R = -jR. It should be obvious at this point that each 90° rotation of the vector multiplies it by j. [A last and final rotation by an additional 90° would give us RE=j4R=j2j2R=(-1)(-1)R = RA and we are now back to our starting orientation.]
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