How i x j= k (vector) BUT ixj = (i)(j) (sin90) = (1)(1)(1) = 1 (Scalar)

Question

How i x j = k (vector) , also in josiah willard Gibbs book who first given the idea of cross product did not explain the mathematical way of cross product. Also from quaternions i found no real conclusion why cross product is vector.

Also i x j = (i)(j)(sin90) (n) why normal vector (n) comes in this equation is there any mathematical explanation to write normal vector here.

Answer 1

For vectors $A$, $B$, the cross product is defined as $$A\times B=||A||\cdot||B||\cdot\sin{(\theta)}\cdot n$$ where $n$ is the vector perpendicular to beth $A$ and $B$. This is why the cross product always results in a vector result.

Answer 2

For quaternionic products, you can also view it as matrix products with the matrices $H^+ (q)$ and $H^-(q)$, as it is shown for example in the book Modélisation quaternionique du lien forme-fonction des surfaces articulaires by Guillard, Hamitouche and Roux, ISBN 9782340010468.

As Peter Foreman said before me, the general equation is: $$C = A\times B=||A||\cdot||B||\cdot\sin{(\theta)}\cdot n$$ With $||A||\cdot||B||$ the norm of $C$ and $n$ is the director unit quaternion. You can find the cross product also via a determinant:

$$A \times B =\begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{bmatrix}$$