If A×B points along positive z-axis then vector a and b lie in
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same axis
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Since AxB is perpendicular to both A and B, they have to lie in the (x,y)-plane if C is to be along z. Also A and B can’t be along the same axis or AxB=0. If C is to be along the positive z axis, the vectors {A,B,C} must be a right hand system, and of course either A or B can’t be 0, but that’s the most you can say. So: any 2 nonzero, noncolinear vectors A and B in the (x,y) plane have AxB along the z-axis, and is along the positive z-axis if {A,B,k} is a right hand system, where k is the unit vector in the z-direction.
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