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By matrix-defined, I mean

 x = | i j k | det(| a b c |) | d e f | 

(instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal)

If I try cross producting two vectors with no k component, I get one with only k, which is expected. But why?

As has been pointed out, I am asking why the algebraic definition lines up with the geometric definition.

By matrix-defined, I mean

$$\left<a,b,c\right>\times\left<d,e,f\right> = \left|

\begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array}

\right|$$

...instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal)

If I try cross producting two vectors with no $k$ component, I get one with only $k$, which is expected. But why?

As has been pointed out, I am asking why the algebraic definition lines up with the geometric definition.