When to us dot versus cross product

[Schmigan]

Hi folks,
When you're squaring the sum of two vectors (v_1 + v_2)^2, why is it that it comes out as v_1 dot v_1 plus 2*v_1 dot v_2 plus v_2 dot v_2? Why do we use the dot product here instead of cross product? I understand that dot product is the multiplication of their parallel components, but it seems arbitrary to use dot rather than cross product in this case

 

[mathwonk]

dot products: v.w = |v| |w| cos(t), hence these are used to measure lengths and angles.

well i guess it isn't so clear from this description, but cross products are used to measure areas, i.e. |vxw| = |v||w| sin(t).

the reason is that one gives the length of the projection of v onto w, and the other gives the length of the projection of v perpendicular to w, hence gives the height needed to compute area of the parallelogram they span.

 

[HallsofIvy]

The cross product of any vector with itself is 0. Not much point in doing that!