As discussed in more detail here, we prefer to define the dot and cross products geometrically, as projections and directed area, respectively. The algebraic formulas are easy to recover from the geometry, but one must first show that both products distribute over addition. This in turn can also be shown geometrically, using the figures below. Each figure can be rotated by dragging with the mouse.
The original Maple figures, which can be found here, were converted to JAVA using JavaViewLib. This HTML5 version was created in 2019 using Geogebra. (The JAVA versions can be found here.)