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 $\displaystyle v_5$ in the same direction as $\displaystyle v_1$ or $\displaystyle v_2$. All you know is that it's perpendicular to $\displaystyle v_1\times v_2$ and $\displaystyle v_3$. Then $\displaystyle v_6=v_5 \times v_4$ is perpendicular to $\displaystyle v_5$ and $\displaystyle v_4$, but we don't know about $\displaystyle v_1$, $\displaystyle v_2$, and $\displaystyle v_3$.

If a vector $\displaystyle w_1$ is perpendicular to $\displaystyle w_2$, and $\displaystyle w_2$ is perpendicular to $\displaystyle w_3$, that doesn't mean that $\displaystyle w_1$ is perpendicular to $\displaystyle w_3$.

If you're confused by all this, try working out an example - say $\displaystyle v_1=i$, $\displaystyle v_2=j$, $\displaystyle v_3=i+j$, and $\displaystyle v_4=k$.

- Hollywood