Question

Prove that if vectors r, s, u, and v in ${\mathrm{ℝ}}^3$ can all be translated to the same plane, thenrole="math" localid="1650041930138" $(r\times s)\times (u\times v)=0$

Short answer

Hence, prove that$(r\times s)\times (u\times v)=0$

Step-by-step solution

Step 1. Given Information

Prove that if vectors r, s, u, and v in ${\mathrm{ℝ}}^3$ can all be translated to the same plane, then$(r\times s)\times (u\times v)=0$

Step 2. As we know that "The cross product of two parallel vectors u and v in ℝ3 is u×v=0.

Hence, $r\times s=0$ when r and sare parallel vectors.

Also $u\times v=0$ when u and vare parallel vectors.

Now we can say that

If r, s, u, and v all lie in some plane P, then the cross products $r\times s$ and $u\times v$ are both orthogonal to P. Therefore, these two vectors are parallel and the cross product of two parallel vectors is $0$.