Dot Products and Components

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Section 11.2 Dot Products and Components

A field can be expressed in many different coordinate systems. For example, in rectangular coordinates, the electric field is

\begin{gather*} \EE = E_x\,\ii + E_y\,\jj + E_z\,\kk . \end{gather*}

Similarly, in cylindrical coordinates,

\begin{gather*} \EE = E_s\,\shat + E_\phi\,\phat + E_z\,\zhat . \end{gather*}

What is the \(x\)-component of \(\EE\text{?}\) Surely just \(E_x\text{.}\) The \(s\)-component? \(E_s\text{.}\)

How could you find these components, given \(\EE\text{?}\) The (scalar) component of a vector field in a given direction is just the projection in that direction. Projections are dot products. Thus,

\begin{gather*} E_x = \EE\cdot\ii \end{gather*}

and

\begin{gather*} E_s = \EE\cdot\shat . \end{gather*}

Given a surface, it makes sense to ask what the component \(E_\perp\) of the electric field is, perpendicular to the surface. By the same reasoning, we have

\begin{gather*} E_\perp = \EE\cdot\nn \end{gather*}

where \(\nn\) is the unit normal to the surface.

The component parallel to the surface, \(E_\parallel\) is more subtle, since there are an infinite number of directions parallel to the surface. One way around this problem is to speaks of vector components, by attaching the direction to the scalar component. For instance

\begin{gather*} \EE_\perp = E_\perp \, \nn . \end{gather*}

We can now define the parallel (vector) component of \(\EE\) as the vector that is left after the perpendicular component is subtracted

\begin{gather*} \EE_\parallel = \EE - \EE_\perp \end{gather*}

from which the magnitude \(E_\parallel=|\EE_\parallel|\) can be computed using the Pythagorean Theorem if desired.