In terms of the standard basis set {i, j, k}, a = 2i - j - 2k, b = 3i - 4 k and c = i - 5 j + 3k.
(i) Find 3a + 2b - 4 c and |a - b|2.
(ii) Find |a|, |b| and a · b. Deduce the angle between a and b.
(iii) Find the component of c in the direction of a and in the direction of b.
(iv) Find a×b, b×c and (a×b)×(b×c).
(v) Find a · (b×c) and (a×b) · c and verify that they are equal. Is the set {a, b, c} rightor left-handed?
(vi) By evaluating each side, verify the identity a×(b×c) = (a · c) b - (a · b) c.