Coordinate Systems

Reasoning about Left and Right Handed Coordinate Systems

Arranging Unit Vectors A coordinate system can be defined by three perpendicular unit vectors. If the coordinate system is Cartesian, which direction does the $+x$ axis point? To resolve this problem, I define an orientation–a coordinate system that is oriented in a certain direction in three-dimensional space. How many unique orientations are there? Here, a unique …

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The Crux of Calculus

Define $\Delta x \equiv x_2 – x_1$, to be consistent with this post. Similarly, define $\Delta y \equiv y_2 – y_1$ and $\Delta z \equiv z_2 – z_1$. The Cartesian coordinates are $x$, $y$, & $z$. In contrast, the spherical coordinates are $r$, $\theta$, & $\phi$. Here, $\phi$ is the azimuthal angle in the $xy$-plane. Next, …

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Spherical Coordinates

The following drawing shows how to convert from Cartesian coordinates to spherical coordinates. This is a short post, but these three equations are pretty useful. I am going to use the end of this post to define the cosine and sine functions: $ \sin\phi \equiv \frac{opp}{hyp}$ $ \cos\phi \equiv \frac{adj}{hyp}$. Here, $ hyp$ is the …

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