`q = q1*q2`

Enter a value for all fields

The Quaternion Multiplication (q = q1 * q2) calculator computes the resulting quaternion (q) from the product of two (q1 and q2).

INSTRUCTIONS: Enter the following:

• (q1):  Enter the scalar(q₄) and i, j and k components (q₁,q₂,q₃) of quaternion one (q1) separated by commas (e.g. 7,4,5,9)
• (q2):  Enter the scalar(q₄) and i, j and k components (q₁,q₂,q₃) of quaternion two (q2) separated by commas (e.g. 7,4,5,9)

Quaternion Multiplication (q): The calculator will return the quaternion that is the product of the two input quaternions.

See full Quaternion Calculator. 

NOTE: Quaternions are not commutative, and the following should be noted:

                   q1*q2 ≠ q2 * q1
                   q1*q2 ≠ -q2 * q1, but
      (q1 * q2) * q3 = q1 * (q2 * q3)

Scalar First Quaternions: Quaternions can be represented in several ways. One of the ways is similar to the way complex numbers are represented:

               q ≡ q₄ + q₁i + q₂j + q₃k,

in which q₁ , q₂ , q₃ and q₄ , are real numbers, and i, j, and k, are unit “vectors” which obey similar rules to the vectors of the same names found in vector analysis, but with an additional similarity to the i of complex arithmetic which equals − 1 . The multiplication rules for i , j , and k are depicted
conceptually as follows:

That is, i j = + k, j k = + i, etc. , from figure 1(a) , and j i = - k, i k = -j , etc., from figure 1(b) . Expressed
in this form, the multiplication rules are very easy to remember. Note that the cross products of i, j , and
k obey the rules of vector cross product multiplication, where, for example, given the orthogonal axes,
x, y, and z: x × y = z, y × z = x , and z × x = y .

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Quaternion Calculators

• Quaternion Addition
• Quaternion Subtraction
• Quaternion Multiplication
• Quaternion Magnitude
• Quaternion Versor
• Quaternion Conjugate
• Quaternion Inverse
• Quaternion of Rotation
• Vector Rotation