Extended exercise on products of four vectors.

(a) Use (4.11) to show

and use (4.13) to simplify the expression on the right-hand side.

(b) Use (4.13) to show that

and show that the right-hand side can be simplified to

(c) Use (4.14) to show that

and simplify the right-hand side further. Note that the product is different from the result in (b), verifying that the position of the brackets matters in cross products.

(d) Use the result in (a) to show that

Take l, m and n to be unit vectors along the sides of a regular tetrahedron. Deduce that the angle between two faces of the tetrahedron is cos^–11/3.

Data from 4.11, 4.13 and 4.14