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. Equivalently, 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, and <,> denotes dot product. The conjugate of q = (s,v) is (s,-v), and the norm of q is the square root of the product (s,v)*(s,-v) = (s*s + ,0) = (s*s + )*h. A unit quaternion is one with norm equal to 1. 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. Show that, for unit vector n, q = (cos(A/2),sin(A/2)*n) is a unit quaternion. 2. Show that the conjugate of a product of quaternions q = (s,v) and q' = (s',v') is the product of the conjugates of the quaterions in reverse order. 3. Use the result of problem 2 to 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 (reciprocal) 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. This reflects commutativity of rotations about a single pole. 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'. Use the result of problem 2. 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. Show that R*n = n.