<s>[INST] plane perpendicular to the straight line I have this problem to find the equation of the plane perpendicular to the straight line joining the points $(1, 3, 5)$ and $(4, 3 ,2)$ at its middle points.  How would you solve this?
 [/INST] The formula for it is: $(X-X_0)\cdot \vec{n}=0$ whereas $X = (x,y,z), X_0 = \left(\frac{1+4}{2}, \frac{3+3}{2}, \frac{2+5}{2}\right)=\left(\frac{5}{2}, 3, \frac{7}{2}\right)$ , and $\vec{n}= \dfrac{1}{\sqrt{3^2+0^2+(-3)^2}}\left(4-1,3-3,2-5\right)=\dfrac{1}{2\sqrt{3}}\left(3,0,-3\right)=\left(\frac{\sqrt{3}}{2}, 0,\dfrac{-\sqrt{3}}{2}\right)$. Can you complete the equation?
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