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Dec2509, 01:10 AM

P: 5

Solving Linear Quaternion Equations
i,j,k, being "righthanded" unit quaternions i',j',k' being the corresponding "lefthanded" unit quaternions q a general right quaternion variable q = w + xi + yj + zk, c = c0 + ic1 + jc2 + kc3 constant right quaternion, solve for q where, 3iq + 5kqj + (2i+k)qi = c ... EQ(1) method, move the known quaternions from R.H.S. to L.H.S of variable q, by changing the "hand" of these quaternions from right to left i.e. qj = j'q^ etc.. then, 3iq^ + 5kj'q^ + (2i+k)i'q^ = c^ ( 3i + 5kj' + 2ii' + ki' )q^ = c^ ... EQ(2) this has the form hq^ = c^ ... EQ(3) with h = ( 3i + 5kj' + 2ii' + ki' ) h*R = ( 3i  5kj'  2ii'  ki' ) h*L = ( 3i  5kj'  2ii'  ki' ) h* = (h*R)*L = (h*L)*R = (  3i' + 5kj' + 2ii' + ki' ) the *R, *L, *, being RightHand Conjugate, LeftHand Conjugate, Total Conjugate resp. first multiply h by its total conjugate on the right to get, hh* = (3i + 5kj' + 2ii' + ki').(3i + 5kj' + 2ii' + ki')* = (3i + 5kj' + 2ii' + ki').(3i + 5kj' + 2ii' + ki') = +39  30jj'  6ji' + 20jk' now take the right conjugate of this, (hh*)*R = ( +39  30jj'  6ji' + 20jk' )*R = ( +39 + 30jj' + 6ji'  20jk' ) and multiply by h* from the left, so, h*(hh*)*R = (3i + 5kj' + 2ii' + ki').( +39 + 30jj' + 6ji'  20jk' ) = ( + 39i  12k  22ii'  130kj' + 21ki' + 195kj' + 20ij' ) Now multiply both sides of EQ(2) by this factor to get, ( + 39i  12k  22ii'  130kj' + 21ki' + 195kj' + 20ij' )( 3i + 5kj' + 2ii' + ki' )q^ = ( + 39i  12k  22ii'  130kj' + 21ki' + 195kj' + 20ij' )c^ i.e. 185q^ = ( + 39i  12k  22ii'  130kj' + 21ki' + 195kj' + 20ij' )c^ Now that we have the real number "185" on the L.H.S of this equation, we can solve for q, q^ = (1/185)( + 39i  12k  22ii'  130kj' + 21ki' + 195kj' + 20ij' )c^ = (1/185)( + 39ic^  12kc^  22ii'c^  130kj'c^ + 21ki'c^ + 195kj'c^ + 20ij'c^ ) So, moving the left handed i',j',k', etc.. factors back over to the right of the constant c, we find, q = ( + 39ic  12kc  22ici  130kcj + 21kci + 195kcj + 20icj ) / 185 this can be verified by pluging the solution back into EQ(1). Curious? This general method is discussed in more detail on PDF page 143 of "Three Quaternion Papers" at archive.org http://www.archive.org/details/Three...nions20062007 


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