Engineer Sphere » Dynamics http://engineersphere.com All the cool kids are doing it. Thu, 08 Jul 2010 03:23:29 +0000 en hourly 1 http://wordpress.org/?v=3.0 Conversions Between Cartesian, Cylindrical and Spherical Coordinates http://engineersphere.com/math/conversions-between-cartesian-cylindrical-and-spherical-coordinates.html http://engineersphere.com/math/conversions-between-cartesian-cylindrical-and-spherical-coordinates.html#comments Sun, 20 Sep 2009 20:58:40 +0000 Luke http://engineersphere.com/?p=877 The 3 common methods of describing a point in a three dimensional coordinate system are Cartesian, Cylindrical and Spherical. The most simple is Cartesian but certain teachers find it necessary to use the others. There are a few simple conversions between them but first it is necessary to know their notation.

Cartesian: (x,y,z)

Cylindrical: (\rho,\phi,z)

Spherical: (r,\theta,\phi)

In most cases you will only need to work from Cartesian to Cylindrical or Spherical OR back, so I will only supply those equations. If you need to work between Cylindrical and Spherical, it would be one more simple step working from one of those, to Cartesian, then on to the other.

Cartesian \rightarrow Cylindrical:

x = \rho \cdot cos (\phi)
y = \rho \cdot sin (\phi)
z = z

Cartesian \leftarrow Cylindrical:

\rho = \sqrt{x^2 + y^2}
\phi = tan^{-1} (\frac{y}{x})
z = z

Cartesian \rightarrow Spherical:

x = r \cdot sin(\theta) \cdot cos(\phi)
y = r \cdot sin(\theta) \cdot sin(\phi)
z = r \cdot cos(\theta)

Cartesian \leftarrow Spherical:

r = \sqrt{x^2 + y^2 + z^2}
\theta = tan^{-1}(\frac{\sqrt{x^2 + y^2}}{z})
\phi = tan^{-1}(\frac{y}{x})

Plug the values from any given points into the correct equation to convert to a different type of coordinate system.

]]> http://engineersphere.com/math/conversions-between-cartesian-cylindrical-and-spherical-coordinates.html/feed 0 Unit Vector Between Two Points http://engineersphere.com/math/unit-vector-between-two-points.html http://engineersphere.com/math/unit-vector-between-two-points.html#comments Sun, 20 Sep 2009 20:17:52 +0000 Luke http://engineersphere.com/?p=858 Suppose you are interested in finding the unit vector between two points, P and Q , which are described in cartesian coordinates as (2,-1,3) and (-1,1,0), respectively.

You would begin by finding the vector between these two points. The direction of this vector may be important so look for key words such as ``from'' P to Q . Once we have established the direction we’re going in, P to Q in this case, we subtract the beginning point from the end point. Q - P . This will give us the vector we are looking for. The next step would be to convert this vector into a unit vector, by dividing it by it’s magnitude.

These are the two formulas we are looking at:
\vec {PQ} = < (Q_x - P_x) , (Q_y - P_y) , (Q_z - P_z) >
\vec U_{PQ} = \frac{\vec {PQ}}{|\vec {PQ}|}

Note:
Here I use \vec {PQ} as my vector from P to Q and \vec U_{PQ} denotes the unit vector from P to Q .

Implementing these formulas:
\vec {PQ} = < (-1 - 2) , (1 - -1) , (0 - 3) > = < -3, 2, -3 >
\vec U_{PQ} = \frac{< -3, 2, -3 >}{|< -3, 2, -3 >|} = \frac{< -3, 2, -3 >}{\sqrt {(-3)^2 + 2^2 + (-3)^2}} = < \frac{-3}{\sqrt 22},\frac{2}{\sqrt 22},\frac{-3}{\sqrt 22}>

Hmm.. The End

]]> http://engineersphere.com/math/unit-vector-between-two-points.html/feed 0 Adding and Subtracting Vectors http://engineersphere.com/math/calculus/adding-and-subtracting-vectors.html http://engineersphere.com/math/calculus/adding-and-subtracting-vectors.html#comments Tue, 08 Sep 2009 04:54:59 +0000 Jeff http://engineersphere.com/?p=797 This is a very simple post and a very simple subject, but every once in a while even the experts need to be reminded how to do simple addition and subtraction with vectors.  Let’s go ahead and specify a couple vectors that we can work with.

Vector A = u_{x} + 2 u_{y} + 3 u_{z}

In MATLAB: >>A = [1 2 4];

Vector B = 2 u_{x} + 3 u_{y} + 4 u_{z}

In MATLAB: >>B = [2 3 4];

When you add vectors together, you add each individual directional component (u_{x}, u_{y}, u_{z} ).  Subtraction works the exact same way.  Lets go ahead and do a few, C will represent the vector that results from the addition and subtraction.

C = A + B = (2 + 1) u_{x} + (2 + 3) u_{y} + (3 + 4) u_{z} = 3 u_{x} + 5 u_{y} + 7 u_{z}

C = A - B = (2 - 1) u_{x} + (2 - 3) u_{y} + (3 - 4) u_{z} = u_{x} - u_{y} - u_{z}

These vectors are all in 3-dimensional space with a X, Y and Z component.  The number in front of each u_{..} directional component is the weight or magnitude of that particular directional component of the vector.  There it is, short and sweet :)

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