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Imagine I have three points $p_1$, $p_2$, $p_3$ in $3$ dimensional space. Imagine I also have two points $c_1$ and $c_2$ in the same space. how to determine if $c_1$ and $c_2$ are on the different sides of $p_1p_2p_3$ plane ?

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Hint: Find the equation of the plane. Plug in $c1$ and $c2$, if the signs match, they’re on same side, else not.

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  • $\begingroup$ How to fin the equation of the plane then. Please answer with the exact formula with x1, x2, x3 and etc. $\endgroup$ – CerushDope Mar 5 '18 at 17:44
  • $\begingroup$ You will need to do some effort as well. You can use vectors, or there is a nice determinant form - both available with a little effort on Google. $\endgroup$ – Macavity Mar 5 '18 at 17:46
  • $\begingroup$ Check #18 at mathworld.wolfram.com/Plane.html for the determinant form, if you can’t find it easily. Try to understand why this works though. $\endgroup$ – Macavity Mar 5 '18 at 17:53
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First you find the plane passing through the three points p1,p2 and p3. It can be done by taking the normal to the plane to be $(p1-p2)\times (p1-p3) = n$. Note that the normal $n$ lies on one of the two sides. Then you find the two vectors connecting the plane to $c1$ and $c2$, i.e. $p1-c1$ and $p1-c2$. At this point the scalar product with the normal indicates how much angle is in between the normal and the two directions. Therefore, if $n\cdot (p1-c1)$ and $n\cdot (p1-c2)$ have different signs, they lie on different sides of the plane.

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HINT

  • find a normal vector $\vec n$ to the plane

  • use dot product by normal and vectors $\vec {PC}$ with $P\in$ plane

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