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October 30th, 2016, 12:36 PM  #1 
Member Joined: Oct 2016 From: Slovenia, Europe Posts: 52 Thanks: 5  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! 
October 30th, 2016, 01:45 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,622 Thanks: 2075 
Have you tried using this formula for the distance from a point to a plane?

October 30th, 2016, 01:48 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,755 Thanks: 695 
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 plusminus 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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