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October 30th, 2016, 12:36 PM   #1
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Question difficult task with vectors, lines and planes

Find all the points on the z axis which lies on the same distance from planes 2x+3y+z=1 and x+y+z=0. I really do not know how to start solving it, although I am familiar with the basic rules and formulas of planes, lines etc. Please help me!
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October 30th, 2016, 01:45 PM   #2
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Have you tried using this formula for the distance from a point to a plane?
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October 30th, 2016, 01:48 PM   #3
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Point on z axis has coordinates (0,0,z).
Line through point perpendicular to 2x+3y+1=1 is has equation (0,0,z)+s1(2,3,1) where $\displaystyle s1=\frac{s}{\sqrt{14}}$.
For plane x+y+z=0, line is (0,0,z)+s2(1,1,1) where $\displaystyle s2=\frac{s}{\sqrt{3}}$.

In the above |s| is the common distance form the point to the planes.

You then have two sets of equations in two unknowns when you set the points from the lines onto the planes.

$\displaystyle s\sqrt{14}+z=1$ and $\displaystyle \pm s\sqrt{3}+z=0$

The plus-minus takes care of the fact that |s| is asked for.

Last edited by mathman; October 30th, 2016 at 04:19 PM. Reason: additional material
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