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Is force or field decomposition into component vectors always valid?

Lets say a constant electric field $\vec{F}$ is acting in space such that it makes an angle $\phi$ with respect to the horizontal direction. The component along x axis (horizontal direction) is $F\cos(\phi)$ and along the vertical $F\sin(\phi)$. Is there any assumption that the space is homogeneous or any other such (uniform space) condition while applying decomposition?

OR is it like decomposition of vectors always refers to a single point in space, and doesn't matter upon the nature of medium/space?

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  • $\begingroup$ That was actually a long historical dispute if you can add vectorial forces together or not. There is a nice educational article on this from EJP or AJP addressing this. Can't find it now, maybe you have a look if you have access to those. $\endgroup$ – mikuszefski Feb 24 '15 at 10:52
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Resolution of force, or any vector in general, into its components along a particular choice of co-ordinate vectors $\{e_i\}$ is not specific to any particular assumption about the medium/space. It is convenient to employ an ortho-normal set of co-ordinate vectors (e.g. ${\hat i} \cdot {\hat j} = 0$ in the Cartesian case), but even if the vectors aren't orthogonal, one can always transform and get to a system which is, see e.g Gram-Schmidt orthogonalization.

You can resolve any vector along another - it is just the dot product ($\vec F \cdot \hat e_j$). But the important issue in such matters is to employ a co-ordinate system that would be useful in the problem. e.g. for a particle's motion on a circle, you can employ a Cartesian co-ordinate system, resolve things along horizontal and vertical, and have all kinds of $\cos \theta$ s and $\sin \theta$ s in your equations. Or you can employ radial co-ordinates, which are naturally more suited to the problem. ($\vec r = r \hat r$, instead of $r \cos \theta \hat i + r sin \theta \hat j$. Beware however, that $\hat r $ changes with $\theta$.)

or in short, resolution of vectors doesn't depend on the ''properties of space'' (I read that as symmetries of the problem). But whether the co-ordinate system you are employing is useful or not, surely depends on these symmetries.

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Such things like as homogeneity or isotropy are important because they are assumed as the basis for the principle of relativity. This principle says you can choose any coordinate system, doesn't matter where it is (since the space is homogeneous) or how is directed (since it is isotropic), or it is still or moves with constant velocity.

So, resolving a vector is just a mathematical problem, but the ability of choosing a coordinate system is a physical problem and surely on the basis of homogeneity and isotropy.

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You are right, the vector decomposition doesn't care about the medium. You decompose your vector into unit length basis vectors in a particular coordinate system, e.g in the Cartesian system,

$$\mathbf{v} = c_1\mathbf{i} + c_2\mathbf{j} + c_3\mathbf{k}$$

This decomposition was done with no reference to any medium or its transformations. In this way you introduce a universal system to break down your vectors and compare them to other vectors.

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