@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.

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@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.

Mathematics
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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?
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either can be initial

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so firsst find your normal line
mmaybe if i show u visually it will be easy for u to do it yourself
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?
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x = -2 is equivalent to -x = 2
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
u can take the cross of these 2 lines to get your new normal then the plane equation is ax+by+cz=d and u can find d but subbing in one of your points
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, dot =0 will also yield the same plane equation
\[u\times v\] gives the normal vector right
yup
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).
pictures r nice :)
Yes, direction of normal vector is not affected by multiplying the vector by a nonzero scalar
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?
both are same, but `7x+y-11z-5=0` looks more neat
14x+2y-2z-10=0 also works
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.
it is an equation, multiplying the same thing both sides doesn't change it in mathematically
Or, whenever I get a negative constant for variable X, I'll just divide it by -1?
why do you care what point you start wid
you end up wid same answer either way right
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.
Just really really making sure
okay i got you, then just multiply -1 in the end to make it look neat :)

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