`q^* = q_4 - q_1i -q_2 j- q_3 k`

Enter a value for all fields

The Quaternion Conjugate (q^* = q₄ ,-q₁, -q₂, -q₃) calculator computes the conjugate (q^*) of a quaternion(q).

INSTRUCTIONS:* Enter the following:

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

Quaternion Conjugate (q*): The calculator returns the quaternion conjugate.

See full Quaternion Calculator. 

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