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3D Coordinate Geometry

Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.

Distance

         

What is the distance between the point \(P=(1,-2,3)\) and the plane \(x-2y+3z-28=0?\)

What is the shortest distance between the point \(P=(-3,3,2)\) and the sphere \(x^2+(y-6)^2+(z+1)^2=1?\)

Find the square of distance between the point \( M_{0}(0, 1, 3) \) and the line \[\frac{ x-4 }{2} = y-7 = \frac{ z+1}{2}. \]

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 \(P ?\)

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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