`q^-1 = (q"*") /|q|^2`

Enter a value for all fields

The Quaternion Inverse (q^-1 = q^*/|q|²) calculator computes the inverse quaternion(q^-1) from the input quaternion (q). 

INSTRUCTIONS: Enter the following:

• (q):  Enter the scalar(q₄) and i, j and k components (q₁,q₂,q₃) separated by commas respectively (e.g. 7,4,5,9) where:
  • q₄ = 7 (scalar) 
  • q₁ = 4 (vector component)
  • q₂ = 5 (vector component)
  • q₃ = 9 (vector component)

Quaternion Inverse (q^-1): The function computes the inverse quaternion.

See full Quaternion Calculator. 

The Math / Science

The formula for the inverse of a quaternion is:

         q^-1 = q^*/|q|²

where:

• q^-1 is the inverse of a quaternion
•  q^* is the conjugate of the quaternion
• |q| is the magnitude of the quaternion

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