## Zenmo one year ago @Small Question, don't need to solve@ Find the general form of the equation of the plane with the given characteristics. Passes through (2, 2, 1) and (-1, 1, -1) and is perpendicular to 2x-3y+z=3.

1. Zenmo

For the points (2, 2, 1) and (-1, 1, -1). How do I know, which point is the initial point? Whichever z variable value is greater?

2. Zenmo

|dw:1435136878321:dw|

3. dan815

either can be initial

4. dan815

5. dan815

mmaybe if i show u visually it will be easy for u to do it yourself

6. Zenmo

So if the solution in the book is 7x+y-11z-5=0, but I put the answer as -7x-y+11z+5=0. I would still be correct?

7. dan815

|dw:1435144221416:dw|

8. ganeshie8

x = -2 is equivalent to -x = 2

9. dan815

you want to find your normal line of the plane, move that normal like to a point, then see the other line connecting p1 and p2, then u have 2 lines and u can define your plane now

10. dan815

u can take the cross of these 2 lines to get your new normal <a,b,c> then the plane equation is ax+by+cz=d and u can find d but subbing in one of your points <x,y,z?

11. dan815

if that plane quations seems a little magical, u can get it more lgoically by computing the dot product of your normal line with one of your vectors in your plane, <x-x1,y-y1,z-z1> <a,b,c> dot <x-x1,y-y1,z-z1> =0 will also yield the same plane equation

12. ganeshie8

$u\times v$ gives the normal vector right

13. dan815

yup

14. Zenmo

ok, I got it now. Thanks for clarification. X = -2 is the same as -X = 2. Which, is the same as u x v = - ( v x u).

15. dan815

pictures r nice :)

16. ganeshie8

Yes, direction of normal vector is not affected by multiplying the vector by a nonzero scalar

17. Zenmo

But, just to make sure. I can put the answer as -7x-y+11z+5=0, even though the solution is 7x+y-11z-5=0?

18. ganeshie8

both are same, but 7x+y-11z-5=0 looks more neat

19. ganeshie8

14x+2y-2z-10=0 also works

20. Zenmo

so to get 7x+y-11z-5=0. I would have to start with the point (2, 2, 1) then. Which, comes back to the question of what initial point to use.

21. ganeshie8

it is an equation, multiplying the same thing both sides doesn't change it in mathematically

22. Zenmo

Or, whenever I get a negative constant for variable X, I'll just divide it by -1?

23. ganeshie8

why do you care what point you start wid

24. ganeshie8

you end up wid same answer either way right

25. Zenmo

Yea, but the signs would be different, hence: -7x-y+11z+5=0 vs 7x+y-11z-5=0. So, either format works, but the positive one looks neater.

26. Zenmo

Just really really making sure

27. ganeshie8

okay i got you, then just multiply -1 in the end to make it look neat :)