Vector cross products of vectors with unique directions.
Suppose we have vectors v1,v2,v3,v4 whose directions are unique. We take the cross product of v1 and v2 which gives us a vector that is not in the direction of v3. If we cross this vector with v3, we will get another vector (call it v5). Will v5 not be in the direction of v1,v2, and v3? Suppose we cross v5 with v4 (whose direction is not the same as v5), will we get a vector (call it v6) whose direction is perpendicular to v1,v2,v3,v4, and v5?
Re: Vector cross products of vectors with unique directions.
You can have in the same direction as or . All you know is that it's perpendicular to and . Then is perpendicular to and , but we don't know about , , and .
If a vector is perpendicular to , and is perpendicular to , that doesn't mean that is perpendicular to .
If you're confused by all this, try working out an example - say , , , and .