Exercise 1.1 What does the condition a (b c) ?0 say about a, b, and c?

Exercise 1.1 The given equation states that a, b, c are not coplanar; a has a nonzero component perpendicular to the plane of b and c, hence a cannot be in this plane.

Exercise 2.1 Show that e ⁱ is determined uniquely by the requirement that x ⁱ = x e ⁱ for every x.

Exercise 2.2 Show that the vectors e ⁱ are linearly independent.

Exercise 2.3 Show that V ?=1/ V.

Exercise 2.4 (a) Show that if the Gram determinant vanishes, then the e _i are linearly dependent, (b) Prove that the Gram determinant equals V ².

Exercise 2.5 (a) Let x= x ^k e _k =x _k e ^k . Write out the modulus of x in all possible forms using the metric tensor, (b) Write out all forms of the dot product x y.

Exercise 2.6 A Cartesian frame is rotated about its third axis to give a new Cartesian frame. Find the matrix of transformation.

Exercise 2.7 Show that .

Exercise 2.8 The contravariant components of