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Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.
What is the distance between the point \(P=(1,-2,3)\) and the plane \(x-2y+3z-28=0?\)
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What is the shortest distance between the point \(P=(-3,3,2)\) and the sphere \(x^2+(y-6)^2+(z+1)^2=1?\)
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Find the square of distance between the point \( M_{0}(0, 1, 3) \) and the line \[\frac{ x-4 }{2} = y-7 = \frac{ z+1}{2}. \]
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If the distance of a point \(P\) other than the origin from the plane \(x+y+z=3\) is equal to the distance of the plane from the origin, which of the following can be the point \(P ?\)
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What is the square of the distance between the two lines \[\begin{align} l&:~x-9=y-8=z-4,\\ m&:~x-10=y-10=z-7? \end{align}\]
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