A quaternion q is an ordered 4-tuple (s,v) = (s,v1,v2,v3) = s*h + v1*i + v2*j + v3*k for vector v = (v1,v2,v3) and unit quaternions h=(1,0,0,0), i=(0,1,0,0), j=(0,0,1,0), k=(0,0,0,1). Multiplication is defined by i*i = j*j = k*k = i*j*k = -h, where h is the multiplicative identity: hq = q for all q. Assuming associativity, it follows that i*j = k, j*k = i, k*i = j, j*i = -k, k*j = -i, and i*k = -j. A unit quaternion is one with s*s +  = 1, where <,> denotes dot product. For a unit vector n, the unit quaternion q = (s,v) = (cos(A/2), sin(A/2)*n) represents an operator that rotates through angle A about axis n: a vector p in 3-space is rotated to p' by computing the quaternion product (0,p') = (s,v)*(0,p)*(s,-v). Note that (s,-v) is the inverse of (s,v): invert by rotating through the same angle about the negative pole. 1. Let q = (s,v) be a unit quaternion. Show that q can be written as (cos(A/2),sin(A/2)*n) for some angle A and unit vector n. 2. Using the above definition of multiplication, prove that the product of quaternions q = (s,v) and q' = (s',v') is q*q' = (s*s'-,s*v' + s'*v + v X v'), where X denotes vector cross product. 3. Show that the norm (squared) of a product of quaternions is the product of norms (squared) of the quaternions so that, in particular, the product of unit quaternions is a unit quaternion. 4. Show that the inverse of a product of quaternions is the product of the inverses in reverse order. 5. Show that (cos(A/2),sin(A/2)*n)*(cos(B/2),sin(B/2)*n) = (cos(A/2+B/2),sin(A/2+B/2)*n) for  = 1. 6. For unit quaternion q, show that q and -q represent the same rotation. 7. State and prove necessary and sufficient conditions for the product of quaternions q and q' to commute. 8. Show that rotation by q' followed by rotation by q is equivalent to rotation by q*q' 9. What rotations (angles and poles) are represented by i, j, k, and i*j*k? 10. For a unit vector n, the rotation matrix represented by q = (cos(A/2),sin(A/2)*n) is R = cos(A)*I + (1-cos(A))*n*n^T + sin(A)*M, where M*p = n X p for all p, and n^T is the transpose of n. a) Show that R*n = n. b) Show that R is orthogonal. Hint: recall that M*M = n*n^T - I.