Which is the best software to make very quick pictures to illustrate math and physics problems (like geometric pitures, cartesian planes, vectors)?
Try using the Python library manim, as described here. 3Blue1Brown is an excellent online mathematics teacher and his videos are very clear! Try watching some of his videos! Python library manin - 3Blue1Brown
I think this depends on your kind of math and physics problems, but
can be used to create graphics that can be exported in vector and pixel formats. If you need a strictly mathematical picture (function plot, definite proportions and distances), GeoGebra is more suitable. If you need something more "artsy", Inkscape will do the job. I sometimes even use both: create something in GeoGebra, export as SVG and change the axis' arrow tips for example.