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 October 7th, 2013, 06:23 AM #1 Senior Member   Joined: Sep 2011 Posts: 140 Thanks: 0 Planes Some quick questions regarding planes: 1) is the plane x+y-z=1 the same as x+y-z=-1 ? 2) also how many planes can there be such that it is orthogonal to the plane Q in R^3 ? thank you
October 7th, 2013, 06:53 AM   #2
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Re: Planes

Quote:
 Originally Posted by gaussrelatz Some quick questions regarding planes: 1) is the plane x+y-z=1 the same as x+y-z=-1 ? 2) also how many planes can there be such that it is orthogonal to the plane Q in R^3 ? thank you
For the first question. The point (1,1,1) lies on the first plane, but not the second, so they are not equivalent.

In fact, (x, y, z) lies on the first plane iff (x, y, z+2) lies on the second. So, the second plane is the first moved 2 units along the z-axis. And, the two planes are parallel.

In answer to the second question, there are an infinite number. If you imagine the x-y plane - defined by z = 0, then any plane defined by x = a or by y =a or by any line in the x-y plane will be othogonal to this.

 October 7th, 2013, 08:16 AM #3 Senior Member   Joined: Sep 2011 Posts: 140 Thanks: 0 Re: Planes Thank you for the quick reply, so even if they are in fact parallel they are indeed different distinct planes yes?? Also I have few more questions: 1) given a point P not on the plane Q, there exists exactly one plane orthogonal to plane Q such that it passes through point P. 2) two distinct lines in R3 always form a unique plane in R3.
October 7th, 2013, 09:26 AM   #4
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Re: Planes

Quote:
 Originally Posted by gaussrelatz 1) given a point P not on the plane Q, there exists exactly one plane orthogonal to plane Q such that it passes through point P.
No. See the example above, where the plane is the x-y plane. Any "vertical" plane is orthogonal to this.

Choose, for example, the point (0, 0, 1). Now, take any line on the x-y plane through the origin. The plane defined by this line projected up the z-axis is orthogonal to the x-y plane and passes through (0,0,1).

These planes are, in fact, defined by y = mx, where m is any constant. And x = 0.

Quote:
 Originally Posted by gaussrelatz 2) two distinct lines in R3 always form a unique plane in R3
There may be no common plane. Consider the line x = 0. All planes that contain that line cut the x-y plane in that line. So, any line that intersects the x-y plane in a point with x not equal to 0 cannot lie on such a plane.

If the lines intersect, then they define a unique plane.

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