PM [B06] Complex Polar Coordinates
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The document summarizes complex polar coordinates, which represents vectors using magnitude, direction, and an imaginary unit i like plane polar coordinates but with complex number notation. It describes that a vector OA is defined by its length r and angle θ as OA = rθ. The x and y components of a vector A are Ax = A cosθ and Ay = iA sinθ, so the vector can be written as A = Acosθ + iAsinθ. While the complex system uses imaginary numbers, the only truly real elements exist along the real polar axis.
![© ABCC Australia 2015 www.new-physics.com
PM [BO5]
THE COMPLEX POLAR COORDINATES](https://image.slidesharecdn.com/pmb06complexpolarcoordinate-151011170738-lva1-app6891/85/PM-B06-Complex-Polar-Coordinates-1-320.jpg)
![© ABCC Australia 2015 www.new-physics.com
THE PARTNER OF COMPLEX VECTORS
To be continued on PM [B07]](https://image.slidesharecdn.com/pmb06complexpolarcoordinate-151011170738-lva1-app6891/85/PM-B06-Complex-Polar-Coordinates-5-320.jpg)
PM [B06] Complex Polar Coordinates
- 1. © ABCC Australia 2015 www.new-physics.com PM [BO5] THE COMPLEX POLAR COORDINATES
- 2. © ABCC Australia 2015 www.new-physics.com Complex Polar Coordinates The complex polar coordinates is similar to the plane polar coordinates in almost all aspects except in notation. The vector OA is also defined by its length 𝑟𝑟 and its angle 𝜃𝜃 as: 𝑶𝑶𝑶𝑶 ≡ 𝑂𝑂𝑂𝑂 𝒓𝒓, 𝜃𝜃 So people just using the system without much minding the differences. 𝜃𝜃 𝑦𝑦 → 𝑖𝑖 𝑥𝑥 → 𝑟𝑟 𝑂𝑂 𝑂𝑂𝐴𝐴 𝒓𝒓, 𝜃𝜃 𝑟𝑟 Pole Polar axis 𝜃𝜃 = 0
- 3. © ABCC Australia 2015 www.new-physics.com The Components The complex components are: 𝑨𝑨𝑥𝑥 = 𝑨𝑨 cos 𝜃𝜃 𝑨𝑨𝑦𝑦 = 𝒊𝒊 𝑨𝑨 sin 𝜃𝜃 The component expression of A: 𝑨𝑨 = 𝑨𝑨 cos 𝜃𝜃 + 𝒊𝒊 𝑨𝑨 sin 𝜃𝜃 The unit vector is: �𝒖𝒖 = cos 𝜃𝜃 + 𝒊𝒊 sin 𝜃𝜃 𝜃𝜃 𝑖𝑖 𝑥𝑥 𝑂𝑂 𝐴𝐴 𝑨𝑨𝑦𝑦=𝑖𝑖𝑨𝑨sin𝜃𝜃 𝑨𝑨𝑥𝑥 = 𝑨𝑨 cos 𝜃𝜃
- 4. © ABCC Australia 2015 www.new-physics.com The Real Values Of course, do not forget that the complex system is a gift from the aliens – it is outlandish and unreal. The only real things you can find is still residing only on the real polar axis. Real axis
- 5. © ABCC Australia 2015 www.new-physics.com THE PARTNER OF COMPLEX VECTORS To be continued on PM [B07]