document.write( "Question 582498: Let the plane V be defined by ax + by + cz + d = 0. Which is true?\r
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Algebra.Com's Answer #372100 by Edwin McCravy(20054) $\"\"$ $\"About$

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document.write( "ax + by + cz + d = 0\r\n" );
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document.write( "Let's find two points in the plane V.\r\n" );
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document.write( "Let y = 0 and z = 0\r\n" );
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document.write( "ax + b(0) + c(0) + d = 0\r\n" );
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document.write( "              ax + d = 0\r\n" );
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document.write( "                  ax = \r\n" );
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document.write( "So (-d/a,0,0) is one point in the plane V\r\n" );
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document.write( "Let x = 0 and z = 0\r\n" );
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document.write( "a(0) + by + c(0) + d = 0\r\n" );
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document.write( "              by + d = 0\r\n" );
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document.write( "                  by = \r\n" );
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document.write( "So (0,,0) is another point in the plane V\r\n" );
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document.write( "The vector between these two points is\r\n" );
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document.write( "⟨0-(),-0,0-0⟩  =  ⟨,,0⟩\r\n" );
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document.write( "This vector ⟨,,0⟩ is parallel to the plane V.\r\n" );
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document.write( "Let's find the dot product of it with ⟨a,b,c⟩\r\n" );
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document.write( "⟨,,0⟩•⟨a,b,c⟩ = ()a + ()b + 0(c) = d-d+0 = 0\r\n" );
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document.write( "Since that dot product is 0, the vector ⟨a,b,c⟩ is perpendicular\r\n" );
document.write( "to to a vector parallel to the plane V and therefore is perpendicular\r\n" );
document.write( "to the plane V.\r\n" );
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document.write( "Therefore ⟨a,b,c⟩ is normal to the plane V since its dot product\r\n" );
document.write( "with a vector parallel to the plane V is 0.\r\n" );
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document.write( "The correct choice is A)\r\n" );
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document.write( "Edwin

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