I mistook your drawings for being 3D representations because you said you wanted to use the cross product.
But if your example is 2D, and V3 represents a 2D normal realtive to those two lines, then your sketches are incomplete, because you're missing a point.
Now this, you can calculate.
If you have three points, V4, V1, and V2, and they form two lines, V4->V1, and V1->V2, and you want to calculate the normal V1->V3, then what you need is not a cross product.
A cross product takes two lines in a plane, and tells you the normal of the plane.
But you have two lines, and you want to determine a line which is at an angle that is half the angle between them.
I'll get back to you on this... I think I know how to do it without using any sin or cosine or arctan.
But if your example is 2D, and V3 represents a 2D normal realtive to those two lines, then your sketches are incomplete, because you're missing a point.
(1) v4----v1 / \ / \ / \ v3 v2 (2) Needs to work for this also v4----v1--------v2 ! ! v3
Now this, you can calculate.
If you have three points, V4, V1, and V2, and they form two lines, V4->V1, and V1->V2, and you want to calculate the normal V1->V3, then what you need is not a cross product.
A cross product takes two lines in a plane, and tells you the normal of the plane.
But you have two lines, and you want to determine a line which is at an angle that is half the angle between them.
I'll get back to you on this... I think I know how to do it without using any sin or cosine or arctan.