document.write( "Question 1046902: prove that the points whos position vectors are 4i j 3k, i 3j 2k and 2i 7k are vertices of a right angle triangle \n" ); document.write( "
Algebra.Com's Answer #662405 by Edwin McCravy(20064)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The signs are missing from the components of the vectors.\r\n" );
document.write( "There are a number of ways the signs could have been so \r\n" );
document.write( "that the claim would be true.  I will choose one way that \r\n" );
document.write( "works.\r\n" );
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document.write( "There are two ways to show that a right triangle is\r\n" );
document.write( "formed by the vectors, by \r\n" );
document.write( "1. showing that the three magnitudes satisfy the Pythagorean \r\n" );
document.write( "theorem \r\n" );
document.write( "2. showing that the dot product of two of the vectors is zero.\r\n" );
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document.write( "I'll use the Pythagorean theorem\r\n" );
document.write( "method.  Let the three vectors be:\r\n" );
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document.write( "A = 4i-j-3k, B = i-3j-2k, and C = 2i-7k  \r\n" );
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document.write( "If vectors P and Q have a common tail, then the\r\n" );
document.write( "vector P-Q can have its tip at the tip of P and\r\n" );
document.write( "its tail at the tip of Q.\r\n" );
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document.write( "Therefore, \r\n" );
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document.write( "the vector that can have its tip at the\r\n" );
document.write( "tip of A and its tail at the tip of B is A-B, and\r\n" );
document.write( "its magnitude is the same as the vector B-A.\r\n" );
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document.write( "A-B = [(4)-(1)]i+[(-1)-(-3)]j+[(-3)-(-2)]k =\r\n" );
document.write( "[4-1]i+[-1+3]j+[-3+2]k = 3i+2j-k\r\n" );
document.write( "∥A-B∥ = ∥B-A∥ = \r\n" );
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document.write( "the vector that can have its tip at the\r\n" );
document.write( "tip of A and its tail at the tip of C is A-C, and\r\n" );
document.write( "its magnitude is the same as the vector A-C.\r\n" );
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document.write( "A-C = [(4)-(2)]i+[(-1)-(0)]j+[(-3)-(-7)]k =\r\n" );
document.write( "[4-2]i+[-1]j+[-3+7]k = 2i-j+4k\r\n" );
document.write( "∥A-C∥ = ∥C-A∥ = \r\n" );
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document.write( "the vector that can have its tip at the\r\n" );
document.write( "tip of B and its tail at the tip of C is B-C, and\r\n" );
document.write( "its magnitude is the same as the vector B-C.\r\n" );
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document.write( "B-C = [(1)-(2)]i+[(-3)-(0)]j+[(-2)-(-7)]k =\r\n" );
document.write( "[1-2]i+[-3]j+[-2+7]k = -i-3j+5k\r\n" );
document.write( "∥B-C∥ = ∥C-B∥ = \r\n" );
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document.write( "By the inverse of the Pythagorean theorem,\r\n" );
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document.write( "∥A-B∥2+∥A-C∥2 =  ∥B-C∥2\r\n" );
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document.write( "So the 3 vectors joining the tips of A,B, and C form\r\n" );
document.write( "a right triangle.\r\n" );
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document.write( "Edwin
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