Math

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)

Calculated Output

w -60

x (i) 12

y (j) 30

z (k) 24

Formula: Hamilton product q₁ ⊗ q₂

Step-by-Step Breakdown

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 this calculator solve your problem today?

Report issue to loopingon.com

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.

Frequently Asked Questions

Is quaternion multiplication commutative?

No — q₁q₂ generally differs from q₂q₁.

The math behind it

Worked Example

Frequently Asked Questions

Related Calculators