I'm not even remotely close to understanding the first problem I dont understand how I can prove this at all. The 2nd problem I really just need help starting If anyone has any inputs at all it would be extremely helpful
The first one is just a matter of substitution and algebra, for example:
$\displaystyle \hat{c}\cdot\hat{b}=c_1b_1+c_2b_2+c_3b_3$
For the second one, you need to find a normal vector. If you label the points A, B, and C (it doesn't matter which is which):
$\displaystyle \hat{n}=(B-A)\times(C-A)$
where $\displaystyle (B-A)$ is the vector from A to B and $\displaystyle (C-A)$ is the vector from A to C. Then the equation of the plane is:
$\displaystyle \hat{n}\cdot(x-A)=0$
Post again in this thread if you're still having trouble.
- Hollywood
Ok, no problem. You just substitute the coordinate notation $\displaystyle (a_1,a_2,a_3)$ for the vector notation $\displaystyle \hat{a}$.
$\displaystyle \hat{c}\times(\hat{a}\times\hat{b})=(\hat{c}\cdot\ hat{b})\hat{a}-(\hat{c}\cdot\hat{a})\hat{b}$
$\displaystyle (c_1,c_2,c_3)\times((a_1,a_2,a_3)\times(b_1,b_2,b_ 3))$ = $\displaystyle ((c_1,c_2,c_3)\cdot(b_1,b_2,b_3))(a_1,a_2,a_3)-((c_1,c_2,c_3)\cdot(a_1,a_2,a_3))(b_1,b_2,b_3)$
$\displaystyle (c_1,c_2,c_3)\times(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1)$ = $\displaystyle (c_1b_1+c_2b_2+c_3b_3)(a_1,a_2,a_3)-(c_1a_1+c_2a_2+c_3a_3)(b_1,b_2,b_3)$
$\displaystyle (c_2(a_1b_2-a_2b_1)-c_3(a_3b_1-a_1b_3),c_3(a_2b_3-a_3b_2)-c_1(a_1b_2-a_2b_1)$,$\displaystyle c_1(a_3b_1-a_1b_3)-c_2(a_2b_3-a_3b_2))$ = $\displaystyle (c_1b_1+c_2b_2+c_3b_3)(a_1,a_2,a_3)-(c_1a_1+c_2a_2+c_3a_3)(b_1,b_2,b_3)$
and I think the rest is pretty clear.
- Hollywood