Hi If I have a general form of a plane such as : $2x + 4y - 6z + 10 = 0$
what is the method for inverting the normal vector?? I realize the normal vector for this equation is $(2, 4, -6)$
would the flip be $(-2, -4, 6)$ perhaps?
thanks
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$\begingroup$Yes negating a vector is "flipping" it, i.e. going along the same line, but pointing backwards.
You can see this because all the vectors on the "vector line" spanned by a vector $v$ have the form $k \cdot v$, for $k \in \mathbb{R}$. If $k > 0$, you just have a larger or bigger vector, and if $k < 0$ you moreover flipped it.
You can get the feeling why that's true that another way. Obviously $(1,0,0)$ flips to $(-1,0,0)$ and so on for $(0,1,0)$ to $(0,-1,0)$ and $(0,0,1)$ to $(0,0,-1)$. But a vector $(a,b,c)$ can be written $$ a\cdot(1,0,0) + b\cdot(0,1,0) + c\cdot(0,0,1). $$ Thus by flipping, we have $$ a\cdot(-1,0,0) + b\cdot(0,-1,0) + c\cdot(0,0,-1), $$ which is equal to $$ (-a,-b,-c) = -(a,b,c). $$
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