The Physical Meaning of the Cross Product and Dot Product 

Loving the humor bits. Just the right amount. And nice editing for those bits, as well. Humor can easily die in a bad edit. But you nailed it. This content also dovetails well with the angular momentum lecture. Including some portion of this as a sidebar might even make that lecture more effective.

Teaching the cross product through torque is a pretty smart way to go about it! Torque is (for the most part) fairly intuitive -- you have to push orthogonally to some lever or bar to rotate it about a pivot point, so that explains why you have y components multiplied by x components and vice-versa. The "minus" seems to come from the fact a rotation can be split into an "up and over" (counter-clockwise) or an "over and down" (clockwise) motion, which requires the moving components be oppositely signed. Still always some frustrating sense of abstraction that seems to linger when we use vectors, but that's hard to avoid. Thanks for another great video!

Good explanation, well done

welcome back boss

The dot product is also used in matrix multiplication. Vector dot products is equal to multiplying a row matrix by a column matrix. For example, ∙ = [[1, 2]] * [[5], [7]] = 5+14=19. Dot products are derived from projections, where proj(a, b) = [(a ∙ b)/(b ∙ b)]*|b|. Cross products, however, comes from the cross-operation sequence. The cross operation involves taking a vector or a group of vectors and outputting a vector that is orthogonal to all vectors being used. For example, a vector in 2D can be crossed to find its perpendicular vector, which proves the perpendicular slope formula, and vectors in 3D can have cross products with 2 vectors, vectors in 4D with 3 vectors, and so on. Area can be interpreted by a cross product of 2 length vectors, as A = bh, with b being a length vector and h being the perpendicular component of the second length vector. Volume can be interpreted by using 3 vectors and using the 4D cross product, as V = Bh, where B is the area of the base, a cross product itself, and h being a perpendicular component of the third vector, so V = Bh = (r ⨉ r)h r ⨉ r ⨉ r (as h = r⊥), but in our 3D world, volume can also mean the DOT product of length and area, due to the box product. Finally, comes the interpretation of cross products in Flatland. We all know that in Flatland, angles exist, so rotations exist. 2D shapes and planar laminae have rotational inertia, so angular momentum and torque exists in Flatland, but since Flatlanders cannot really see the objects rotating due to a 1D vision, they usually don't think about torque, as the torque will be bending into the 3rd dimension. We 3D beings can see objects rotate about an axis, but we cannot interpret solid angular motion. This is because solid angular momentum is changed by 3-torque, which is equal to r ⨉ A ⨉ F, which goes into the 4th dimension. However, 4D beings can comprehend solid angular velocity and objects rotating about a plane rather than an axis. Finally, comes the 2nd moment of area, which is equal to A ⨉ A, or (r ⨉ r) ⨉ (r ⨉ r). This requires 6 dimensions, as the first cross product gives 3 dimensions, and the second gives 3 more dimensions.

Such a fun video 🎉

muy buen video, muchisimas gracias

Is there like a book that would help learn k-12 mathematics conceptually instead of rote memorization from government schooling?

Instead of the cross product you should really use the geometric product and the bivectors from the geometric algebra. Also bivector lies in the plain of the rotation, not in some random axis.