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I think of quaternions as operators which are applied to co-ordinates of a point. Operator $i$ rotates the point, to which it is applied, by 90 degrees anti-clockwise about positive $z$ axis. Operator $j$ rotates the point by 90 degrees about positive $y$ axis. Operator $k$ rotates the point 90 degrees about positive $x$ axis.

So, $i\cdot j=k$ actually means $j(i(x,y,z))=k(x,y,z)$, where $x, y$ and$z$ can be the co-ordinates of any arbitrary point.

So, each quaternion must contain the information about a particular axis and a particular angle to rotate the point on which it's applied. I think that's why they have four components. Three components are a must to describe any particular axis and one component to specify the angle.

$q_1\cdot q_2=q_3$ means if we rotate a point $(x,y,z)$ by the angle specified by $q_1$ about the axis specified by $q_1$, then rotate the resulting point by the angle and axis specified by $q_2$, then the point we end up is the same as the point we'd have ended up if we we'd rotated the initial point by the angle and axis specified by $q_3$.

Does this interpretation seem correct?

If this is correct, then what is the angle and axis encoded in an arbitrary quaternion $a+bi+cj+dk$?

Also, I have a problem with this interpretation. Because $i\cdot i=j \cdot j=k \cdot k=-1$. But $i\cdot i(x,y,z) = (-x,-y,z)$, which is not equal to $j\cdot j(x,y,z)=(-x,y,-z)$ and $k\cdot k(x,y,z)=(x,-y,-z)$

Ryder Rude
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  • Maybe useful: Quaternions and spatial rotation. – Mauro ALLEGRANZA Dec 11 '17 at 14:06
  • See (for example) this recent question: https://math.stackexchange.com/questions/2560495/quaternions-rotations-and-real-numbers (and the many questions appearing in the "Related" column there). – Hans Lundmark Dec 11 '17 at 15:35
  • @HansLundmark I now get that $q(x,y,z)=q(xi+yj+zk)q'$ which is a rotation of the point $(x,y,z)$ by the ange and axis specified by $q$. So, this gives a physical meaning to the product $q(xi+y+zk)q'$. – Ryder Rude Dec 12 '17 at 02:29
  • @HansLundmark I need a physical meaning of the product $q_1\cdot q_2$ of any two quaternions. Does $q_1 \cdot q_2$ represent the resultant axis and resultant angle of the individual axes and angles encoded in $q_1$ and $q_2$? If not, then what else does it represent? – Ryder Rude Dec 12 '17 at 02:32
  • Insofar as quaternions represent rotations, the product $q_1q_2$ represents the rotation that is a composition of the two rotations associated to $q_2$ and $q_1$. (Note that nonzero rotations can be used to represent 3D rotations, but then the quaternions $q$ and $-q$ always represent the same rotation. Also, 4D rotations can be represented, but they are represented by pairs of quaternions, and again $(p,q)$ and $(-p,-q)$ represent the same 4D rotation. Ultimately, quaternions represent spin, which is a bit more nuanced than just rotation.) – anon Dec 12 '17 at 02:33

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I detected a problem with my interpretation. I thought that just like $i$ in complex numbers represented 90 degree rotation about z axis, so it must be similar with quaternions. But it's different here.

In quaternions, $i, j,$ and $k$ represent 180 degree rotations about x, y and z axes respectively.

So, $j(i(x,y,z))=j(x,-y,-z)=(-x,-y,z)$ and $k(x,y,z)$ is 180 degree rotation of $(x,y,z)$ about z axis, therefore, $k(x,y,z)=(-x,-y,z)$. So, $j(i(x,y,z))=k(x,y,z)$ which means $i\cdot j=k$.

Also, $i(i(x,y,z))=j(j(x,y,z))=k(k(x,y,z))=(x,y,z)$ because they all represent 360 degree rotation of the point $(x,y,z)$. So, $i\cdot i= j\cdot j=k\cdot k$.

Ryder Rude
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