Transcription of Introducing The Quaternions
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Introducing The Quaternions
Introducing The QuaternionsIntroducing The QuaternionsJohn HuertaDepartment of MathematicsUC RiversideFullerton CollegeIntroducing The QuaternionsThe Complex NumbersIThe complex numbersCform a operations are very related to particular, multiplication by a unit complex number:|z|2=1which can all be written:z=ei gives arotation:Rz(w) =zwby angle . Introducing The QuaternionsThe Complex NumbersHow does this work?IC={a+bi:a,b R,i2= 1}IAny complex number has a length, given by thePythagorean formula:|a+bi|= a2+ can add and subtract inC. For example:a+bi+c+di= (a+c) + (b+d) can also multiply, which is much messier:(a+bi)(c+di) = (ac bd) + (ad+bc)iWhat does this last formula mean? Introducing The QuaternionsThe Complex NumbersFortunately, there is a better way to multiply complex numbers,thanks to Leonhard euler :Figure: Handman s portrait of proved:ei =cos +isin Introducing The QuaternionsThe Complex NumbersGeometrically, this formula saysei lies on the unit circle inC:Figure: euler s The QuaternionsThe Complex NumbersIei has unit we multiply by a positive number,r, we get a complexnumber of lengthr:rei.
Rotations Using Quaternions But there are many more unit quaternions than these! I i, j, and k are just three special unit imaginary quaternions. I Take any unit imaginary quaternion, u = u1i +u2j +u3k. That is, any unit vector. I Then cos’+usin’ is a unit quaternion. I By analogy with Euler’s formula, we write this as: eu’:
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