[imh]

A scalar is just a magnitude (call it amount). A vector is a direction with magnitude (call it length). A bivector is a pair of directions with magnitude (call it area). You can keep going and make something with volume and even more into higher dimensions.

Imagine putting a 2d rectangle into 3d space with some orientation. Starting from some corner of the rectangle, you have two sides coming out from it. In 3d space, those sides make 3d vectors. You can generalize it to 3 vectors making a cube with magnitude (volume). And these don't have to be perfect rectangles and cubes, they can be parallelograms and parallelopipeds (3d parallelograms) and higher dimensional analogues.