October 7th, 2013, 07:23 AM  #1 
Senior Member Joined: Sep 2011 Posts: 140 Thanks: 0  Planes
Some quick questions regarding planes: 1) is the plane x+yz=1 the same as x+yz=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, 07:53 AM  #2  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,312 Thanks: 115  Re: Planes Quote:
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 zaxis. And, the two planes are parallel. In answer to the second question, there are an infinite number. If you imagine the xy plane  defined by z = 0, then any plane defined by x = a or by y =a or by any line in the xy plane will be othogonal to this.  
October 7th, 2013, 09:16 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, 10:26 AM  #4  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,312 Thanks: 115  Re: Planes Quote:
Choose, for example, the point (0, 0, 1). Now, take any line on the xy plane through the origin. The plane defined by this line projected up the zaxis is orthogonal to the xy 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:
If the lines intersect, then they define a unique plane.  

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