Q. 79 - Calculus

Question

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-by-Step Solution

Answer

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

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 ℝ 3 is u × v = 0 .

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

Also $u\times v=0$ when u and v are 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 $0$.