Quaternion Calculator

Multiply two quaternions (Hamilton product).

q₁ w

q₁ x (i)

q₁ y (j)

q₁ z (k)

q₂ w

q₂ x (i)

q₂ y (j)

q₂ z (k)

w -60

x (i) 12

y (j) 30

z (k) 24

Formula: Hamilton product q₁ ⊗ q₂

Step-by-step with your numbers:

1. Values used:

2. q₁ w = 1

3. q₁ x = 2 i

4. q₁ y = 3 j

5. q₁ z = 4 k

6. q₂ w = 5

7. q₂ x = 6 i

8. q₂ y = 7 j

9. q₂ z = 8 k

10.

11. w = -60

12. x = 12i

13. y = 30j

14. z = 24k

Did we solve your problem today?

Quaternions extend complex numbers to four dimensions and are used for 3D rotations in graphics and robotics.

The math behind it

The Hamilton product is non-commutative. It combines the scalar parts and the cross/dot products of the vector parts following i² = j² = k² = ijk = −1.

Worked example

(1 + 2i + 3j + 4k)(5 + 6i + 7j + 8k) = −60 + 12i + 30j + 24k.

FAQ

Is quaternion multiplication commutative?

No — q₁q₂ generally differs from q₂q₁.

The math behind it

Worked example

FAQ

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