By considering the cross products of the usual unit vectors <1,0,0>, <0,1,0> and <0,0,1>, it was shown that the cross product is not commutative! In fact the direction of the cross product is determined by the right hand rule.
Tensor Notation was introduced, because I think this makes the proofs of vector identities less tedious. However, the textbook does not use tensor notation, and proves vector identities the tedious way (by expanding and comparing components). You can find a good introduction to tensor notation in the following reference:
The Kronecker Delta and the Anti-symmetric Tensor were introduced, which allowed us to express the dot-product and the cross-product. Tensor notation was then used to prove that the direction of a cross-product is perpendicular to both original vectors.
We finished the lecture by listing the six main vector identities. These can be proved easily using tensor notation, and in fact, a couple of these proofs will be shown next lecture.