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, in mathematics, are a non-commutative extension of complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations, such as in 3D computer graphics. Since the affine transformations representable by matrices are a superset of quaternion rotations, matrices are employed as a computationally suboptimal but geometrically equivalent orientation representation in many calculations. Vectors cannot represent rotations per se, only directions, and so are generally unsuitable unless the calculation in question assumes a fixed "up" direction.
In modern language, quaternions form a 4-dimensional normed division algebra over the real numbers. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by (Unicode ℍ). It can also be given by the Clifford algebra classifications Cℓ0,2(R) = Cℓ03,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only three finite-dimensional division rings containing the real numbers as a subring.Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = −k, ij = −ji
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How you represent rgb color by quaternion in matlab?
we can represente a RGB color in matlab by a pure quaternion q=0+R*I+G*J+B*K we apply the function q=quaternion(0,R,G,B); for this you must download the qtfm toolbox.
Can you add speed with velocity?
Yes, speed is a scalar and velocity is a vector, adding them together is called a quaternion or complex motion (s + v) = [s,v]. Complex numbers in geometry are 2 dimensional quaternion subsgroups.
What is a vector quatinty?
A vector is a part of a more general number or quantity, called a Matrix or Quaternion. Vectors were developed by William Rowan Hamilton as part of a Quaternion. The Quaternion consists of a scalar part 1 and three vectors I ,J and K.. The scalar is a real number and InJ and K are vector numbers, where I^2 = J^2 = K^2 = -1.. Vectors denote directions such as an axis in space, Ix + Jy + Kz. Quaternions Q = s + V = Q(cos(Q) + Vsin(Q)) = [cos(Q), sin(Q)(I + J + K)] . The Quaternion is vector if the angle is an even multiple 90 degrees and the Quaternion is a vector when the angle is a odd multiple of 90 degrees.. If the angle is not a multiple of 90 degrees, the Quaternion is part scalar and part vector. Most variables in physics are Quaternions. ============================================ A 'vector' quantity is a quantity that has both a size and a direction. Examples are: force, velocity, acceleration, and electric field. A 'scalar' quantity is a quantity that has size but no direction. Examples are: cost, temperature, speed, and volume.
Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
What is the role of mathematics in the field of physics?
The Laws of Physics are mathematical and mathematics allows one to read and understand the Physical Laws.Before Newton, mathematics was seen as a tool for Physics. Now mathematics is the microscope and telescope for physics.The Universe consists of four dimensions (quaternions) and Quaternion Mathematics is rarely known among physicists and seldom taught.Consequently, much of the mysteries of "dark" physics is the result of lack of knowledge of Quaternion Mathematics.
