Question

1. The quaternions H are a set of "hyper-complex" numbers of the form p = a + bi-cit dk. where a, b, c, d E R. Like the complex numbers, they can be added, conjugated by sending p ? p = a-bi-cj-dk, and the norm of a quaternion is given by lp-va2? 2 247. To multiply two quaternions, we use the algebra i2 = j2 = k2 =-1 ij=-ji = k jk=-kj = i ki=-ik = j a) Use the Pauli matrices ? , ???, to find a representation of the quaternions. b) An maginary quaternion (a = 0) corresponds to a 3-vector p = p,i + pyJ +Pak. Show that multiplication of two imaginary quaternions p and q contains the ordinary dot and cross product. c) The set of unit quaternions psuch that p defines the three-sphere S3, just like the coinplex numbers of unit norm define the unit circle. Describe the image of the map p ? pip, as well as the preimage of a single point

Answer 1

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