vector plot with magnitude as legend

Question

Mathematica does provide beautiful visualization of vectors.

However, one puzzling situation I'm always facing is how to visualize the magnitude of vector fields. I'd like to have a plot as shown in the figure below with a box showing the magnitude of a reference vector.

How can I plot vector fields with legends showing the reference magnitude?

Answer 1

If I understand your question correctly, your first idea for magnitude visualization is the comparison to a reference vector. The reason for your idea to not be realized in mathematica adhoc, is that there are other ways that are more vivid in showing magnitude of a vector field. Color is the keyword. There are two possibilities I can quickly think of, to implement the magnitude information via colors.

1. Vectors with colors

There is an option for the command VectorPlot called VectorColorFunction which defines a color palette representing the vector magnitude. This is by default scaled, which can be changed with the command VectorColorFunctionScaling. You can then use for example BarLegend to create a bar legend for your color palette. You will have to give the data range of your magnitudes. Below you can see an example of this.

With the code for this plot:

VectorPlot[{Exp[x], Exp[y]}, {x, -1, 1}, {y, -1, 1},
VectorColorFunction -> "Rainbow", 
PlotLegends -> BarLegend[{"Rainbow", {0.52026, 3.84423}}]]

Note, that I chose the range values in BarLegend manually as i checked the minimum and maximum values.

2. Coloring the background

There is another function for vector field illustration, called VectorDensityPlot. It works similar to VectorPlot. Just have a look at the options of this command and you will easily come to something like this: