One vector carrying two facts at once: which plane a pair of arrows spans, and how much area they enclose in it.
Swap a and b — mentally, or by moving them past each other — and the cross product points the other way.
Drag to orbit; slide the components; add the third vector.
Two vectors span a parallelogram. The cross product packages that parallelogram into a single vector: it stands perpendicular to the plane the two of them span, and its length is the parallelogram's area. One object carrying both the orientation of the plane and the size of the patch.
Which way perpendicular? There are two choices and no way to decide from the vectors alone, so we simply agree: curl the fingers of your right hand from a to b and your thumb points along a × b. That convention is the reason cross products anti-commute — going b to a curls the other way, so b × a = −(a × b).
Slide the two vectors toward each other and watch the area shrink to nothing: parallel vectors span no parallelogram, so their cross product is the zero vector. That is the exact mirror of the dot product vanishing for perpendicular vectors.
Turn on the third vector and the parallelogram becomes a box. Its volume is (a × b) · c — area of the base, times the height, and the height is what the dot product measures. That number is also the determinant of the three vectors written as a matrix, which is where the determinant's "how much does this transformation scale volume" meaning comes from, and eventually the Jacobian.