The second example [Transparancies 2 and 3] involved angular velocity, which is defined in terms of a cross product.
As promised last lecture, a proof was given of the distributivity of the cross product over addition [Transparancies 3 and 4]. This proof differs from the geometrical one given in the textbook on pages 47-48. Instead, we defined the cross product in terms of the "anti-symmetric tensor", and used "Einstein Summation Notation" (implicitly summing over repeated indices). This made the proof quite easy. (More on tensor notation can be found in section 1.15 of the textbook.)
Using the dot product and cross product, we looked at the decomposition of a vector into components parallel to and perpendicular to another vector [Transparancies 5 and 6].
The "Triple Scalar Product" was defined [Transparancy 6]. Using the anti-symmetric tensor, it was shown that the dot and cross can be interchanged, and in fact the triple scalar product can be written as the determinant of a matrix.
A geometrical interpretation of the triple scalar product was given [Transparancy 7]. It turns out to be the volume of a parallelepiped.
Here are the scanned-in transparancies, in full-colour JPEG format:
