Find a vector parallel to two planes

AI Thread Summary
To find a vector parallel to the two planes defined by the equations 3x+y+z=1 and 3x+z=0, the normal vectors are identified as <3,1,1> and <3,0,1>. The cross product of these normal vectors yields a vector that is perpendicular to them, thus parallel to the planes. The calculated vector <1,0,-3> is valid, and other multiples like <2,0,-6> are also acceptable solutions. Multiple correct answers exist for this problem, confirming that various scalar multiples can represent the same direction.
UrbanXrisis
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there are two planes:

3x+y+z=1
3x+z=0

find a vector U with positive first coordinate that is parallel to both planes.

the way I went about solving this:
the normal vectors are: <3,1,1> and <3,0,1>, the cross product of the vectors will give a vector perpendicular to the normal, which means it would be parallel to the two planes right?
 
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Yes, that's right.
 
when I did this, I get <1,0,-3> and this is not correct. so i do not know where I went wrong
 
(1, 0, -3) is correct. Maybe you are comparing it to a vector that is a multiple of (1, 0, -3)--there is more than one answer. (2, 0, -6) is also correct.
 

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