Stokes Theorm

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October 27th 2012, 03:00 AM
mrmaaza123

Stokes Theorm

I am attempting to solve a question a question of stokes theorm, but i have got stuck with this question in which i have to first find the equation of a line passing through three points.

The points are (3,0,0),(0,3/2,0),(0,0,3).
I remember doing something like this in calculus class but i dont remember how exactly to go about such problems.

Someone please help me out with this.

Help will be highly appreciated.
October 27th 2012, 03:46 AM
johnsomeone

Re: Stokes Theorm

Two distinct points determine a line. When you have three distinct points, they *might* all be on the same line (in which case those points would be called "collinear"). So, do it for two points, then see if the 3rd point is also on that line.
So, pick two of those points (say (3, 0 0) and (0, 0, 3) since their the simpliest) and find the line between them. There are a few ways to do this. I prefer vectors.

$\text{Let } \vec{v} = 3\hat{i} - 3\hat{k} \ ( \ = (3, 0, 0) - (0, 0, 3) = (3, 0, -3) \ ).$

$\text{Let } \vec{w} = 3\hat{i} \ ( \ = (3, 0, 0) \ ).$

$\text{Then } \vec{L}(t) = \vec{v} \ t + \vec{w} \text{ is the equation of the line through those two points.}$

$\text{Note that } \vec{L}(t) = \vec{v} \ t + \vec{w} = (3\hat{i} - 3\hat{k}) \ t + (3\hat{i}) \in \text{LinearSpan} \{ \hat{i}, \hat{k} \},$

$\text{and so } \vec{L}(t) \ne (3/2)\hat{j} \text{ for any value of } t.$

Thus those three points are not collinear - there is no single line going through all three of them.
October 27th 2012, 06:20 AM
mrmaaza123

Re: Stokes Theorm

Ok i got that , but what if i want to find the equation for the triangle that passes through these three points ? How do i go about that ?
October 27th 2012, 09:48 AM
johnsomeone

Re: Stokes Theorm

"The equation of a triangle"? What do you mean?
1) the equations for the 3 line segments that comprise the triangle (3 different equations, since the points are not collinear).
2) the equation of the plane passing through those three points.
3) the formula for any point in the convex hull formed by those three points (i.e. all points in the filled-in triangle).
October 27th 2012, 05:47 PM
mrmaaza123

Re: Stokes Theorm

Sorry for not being clear, but what i meant was , how do i find the equation of the plane passing through these three points, which actually is a tringle in the 3D space ?

Is it going to be x+2y+z=3 ?
October 27th 2012, 10:55 PM
johnsomeone

Re: Stokes Theorm

That's correct. The equation of the plane containing those 3 points is x+2y+z = 3.
Since three distinct non-collinear points determines a unique plane, and since those three points each satisfy that equation for a plane, you can be sure that that equation is the equation of the plane containing those three points.