How can this indicate rotation?

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In summary, the equations r2= (x-a)2 + (y-b)2 + (z-c)2 and r2= (x- ia)2 + (y- ib)2 + (z- ic)2 can represent displacement and rotation, respectively. This is due to the fact that i, representing a rotation of 90 degrees, can be used to represent 2-dimensional coordinates, and quaternions can be used to represent 3-dimensional coordinates and perform rotations.
  • #1
yicong2011
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If we say

r2= (x-a)2 + (y-b)2 + (z-c)2

can generally represent displacement,

why can

r2= (x- ia)2 + (y- ib)2 + (z- ic)2

generally represent rotation?

(i2 = -1)


And why

r2= x2 + y2 + (z - ia)2

represents rotation with respect to z axis
 
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  • #2
I'm not quite sure what you're saying here.

What I see is the equation of a sphere of which the center has been translated.
First over real coordinates, and then over imaginary coordinates.

What I can tell you, is that if you can for instance represent 2-dimensional coordinates by a number in the imaginary plane, that multiplication by i equals rotation over an angle of 90 degrees.

More generally if you have 3-dimensional coordinates, you can represent them by a number in quaternion space. Multiplication with an other quaternion number comes out as a rotation.
 

1. What is rotation and why is it important to study?

Rotation is the movement of an object around an axis or center point. It is important to study because it is a fundamental concept in physics that helps us understand the motion of objects and phenomena in our world.

2. How can we measure rotation?

Rotation can be measured in different ways depending on the object and situation. Some common methods include using a protractor or angle-measuring tool, observing changes in position over time, and using specialized equipment like gyroscopes or accelerometers.

3. What are some examples of rotation in everyday life?

Some examples of rotation in everyday life include the rotation of the Earth around its axis, the rotation of a bicycle wheel, and the rotation of a ceiling fan. Other examples include the rotation of a spinning top, the rotation of a car tire, and the rotation of a record on a turntable.

4. How can we determine the direction of rotation?

The direction of rotation can be determined by using the right-hand rule, which states that if you point your right thumb in the direction of the rotation, your fingers will curl in the direction of the rotation. Another way to determine the direction of rotation is by observing the direction of the object's motion in relation to its axis or center point.

5. What are some factors that can affect rotation?

Some factors that can affect rotation include the speed of the object, the shape and size of the object, the presence of external forces like friction or air resistance, and the object's center of mass. The type of axis or center point around which the object is rotating can also affect its rotation.

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