Vector Projection; Space Coordinate System

Why is it that we have to provide a reference point $P(x, y, z)$ to convert a Cartesian vector to a Cylindrical vector? Also, how would you interpret the vector $A_\rho a_\rho + A_\phi a_\phi + A_z a_z $, geometrically?

#Physics #Engineering #Math

Easy Math Editor

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Comments

Cylindrical polar coordinates are defined in terms of an axis (the axis of the cylinder). The relationship between the fundamental Cartesian coordinate vectors $a_x,a_y,a_z$ (which are unit vectors pointing along the $x$ -, $y$ - and $z$ -axes) and the fundamental cylindrical polar coordinate vectors $a_\rho,a_\phi,a_z$ varies depending on the point where these vectors are being considered. While, for example, $a_x$ always points in the same direction, the vector $a_\rho$ points in different directions at different points in space.

To be specific, if $a_x=(1,0,0)$ , $a_y = (0,1,0)$ and $a_z = (0,0,1)$ . Suppose we are at the point $P$ with coordinates $(\rho\cos\phi,\rho\sin\phi,z)$ .

$a_\rho$ is the unit vector in the "direction of increasing $\rho$ ", namely the unit vector which is perpendicular to the $z$ -axis and radially outwards, so $a_\rho = (\cos\phi,\sin\phi,0)$ .
$a_\phi$ is the unit vector in the "direction of increasing $\phi$ ", namely the direction you would go if $\phi$ increases, while keeping $\rho$ and $z$ constant. If you keep $\rho$ and $z$ constant, you rotate about the $z$ -axis, and so $a_\phi = (-\sin\phi,\cos\phi,0)$ .
$a_z$ is still the unit direction in the "direction of increasing $z$ ", namely $(0,0,1)$ .

If we are converting between the Cartesian and cylindrical polar systems, we are in the business of describing the same vector $a$ in two different ways, and so $a \; = \; A_xa_x + A_ya_y + A_za_z \; = \; A_\rho a_\rho + A_\phi a_\phi + A_z a_z$ Taking scalar products: $\begin{array}{rcl} A_\rho & = & a \cdot a_\rho \; = \; A_x \cos\phi + A_y \sin\phi \\ A_\phi & = & a \cdot a_\phi \; = \; -A_x \sin\phi + A_y \cos\phi \\ A_z & = & a \cdot a_z \; = \; A_z \end{array}$ The last equation is trivial, since the $z$ -coordinate is the same coordinate in both coordinate systems. These are precisely the equations that are being used in the example you provided.

Mark Hennings - 7 years, 9 months ago

Thank you Sir Mark :))

JohnDonnie Celestre - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$