Quaternions: The skew field H of quaternions can be identified with the vector space R4 equipped with a noncornmutative multiplication operation. The standard basis vectors e1, e2, e3, e4 are traditionally denoted by the letters 1, i, j, k; the vector (a, b, c, d)T ∈ R4 corresponds to the quaternion q = a + bi +cj + dk. Quaternion addition coincides with vector addition. Quaternion multiplication is defined so that
1q = q = q1,
i2 = j2 = k2 = -1,
i j = k = -j i,
i k = -j = - k i,
j k = i = -k j.
along with the distributive laws
(q + r)s = qs + rs, q (r + s) = qr + qs,
for all q, r, s ∈ H.
(a) Compute the following quaternion products:
(i) j(2 - 3j + k)
(ii) (1 + i)(1 - 2i + j)
(iii) (1 + i - j - 3k)2
(iv) (2 + 2 i + 3 j - k )(2 - 2 i - 3 j + k)
(b) The conjugate of the quaternion q = a + b i + cj +d k is defined to be  = a - bi - cj - d k. Prove that q = ||q||2 = q, where ||∙|| is the usual Euclidean norm on R4.
(c) Prove that quaternion multiplication is associative.
(d) Let q = a + bi + cj + d k ∈ H be a fixed quaternion. Show that Lq[r] = qr and Rq [r] = rq define linear transformations on the vector space E ≃ R4. Write down their 4 × 4 matrix representatives, and observe that they are not the same since quaternion multiplication is not commutative.
(e) Show that Lq and Rq are orthogonal matrices if ||q||2 = a2 + b2 + c2 + d2 = 1.
(f) We can identify a quaternion q = b i + c j + d k with zero real part, a = 0, with a vector q = (b, c, d)T ∈ R3. Show that, in this case, the quaternion product qr = q × r - q ∙ r can be identified with the difference between the cross and dot product of the two vectors. Which vector identities result from the associativity of quaternion multiplication?