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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, which are excerpted from the reference above.
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If you have Maple, clicking on each figure should open a Maple worksheet containing a copy of the figure, which can be rotated in 3 dimensions.