Equation of a Plane and Proofs

**Quixomatic** · April 20th 2010, 06:20 AM

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

**hollywood** · April 21st 2010, 05:43 AM

The first one is just a matter of substitution and algebra, for example:

$\hat{c}\cdot\hat{b}=c_1b_1+c_2b_2+c_3b_3$

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):

$\hat{n}=(B-A)\times(C-A)$

where $(B-A)$ is the vector from A to B and $(C-A)$ is the vector from A to C. Then the equation of the plane is:

$\hat{n}\cdot(x-A)=0$

Post again in this thread if you're still having trouble.

- Hollywood

**Quixomatic** · April 22nd 2010, 04:54 AM

I still dont understand that first problem where you have to prove the equation

**hollywood** · April 22nd 2010, 07:17 PM

Ok, no problem. You just substitute the coordinate notation $(a_1,a_2,a_3)$ for the vector notation $\hat{a}$ .

$\hat{c}\times(\hat{a}\times\hat{b})=(\hat{c}\cdot\ hat{b})\hat{a}-(\hat{c}\cdot\hat{a})\hat{b}$

$(c_1,c_2,c_3)\times((a_1,a_2,a_3)\times(b_1,b_2,b_ 3))$ = $((c_1,c_2,c_3)\cdot(b_1,b_2,b_3))(a_1,a_2,a_3)-((c_1,c_2,c_3)\cdot(a_1,a_2,a_3))(b_1,b_2,b_3)$

$(c_1,c_2,c_3)\times(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1)$ = $(c_1b_1+c_2b_2+c_3b_3)(a_1,a_2,a_3)-(c_1a_1+c_2a_2+c_3a_3)(b_1,b_2,b_3)$

$(c_2(a_1b_2-a_2b_1)-c_3(a_3b_1-a_1b_3),c_3(a_2b_3-a_3b_2)-c_1(a_1b_2-a_2b_1)$ , $c_1(a_3b_1-a_1b_3)-c_2(a_2b_3-a_3b_2))$ = $(c_1b_1+c_2b_2+c_3b_3)(a_1,a_2,a_3)-(c_1a_1+c_2a_2+c_3a_3)(b_1,b_2,b_3)$

and I think the rest is pretty clear.

- Hollywood

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