document.write( "Question 1203666: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and only if ||u + v|| = ||u − v||. \n" ); document.write( "

Algebra.Com's Answer #839412 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "(a) Show that 4u · v = ||u + v||^2 − ||u − v||^2.
\n" ); document.write( "(b) Show that u and v $\"highlight%28cross%28is%29%29\"$ $\"highlight%28highlight%28are%29%29\"$ orthogonal if and only if ||u + v|| = ||u − v||.
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document.write( "             Part (a)\r\n" );
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document.write( "    ||u + v||^2 = (u,u) + 2(u,v) + (v,v)      (1)\r\n" );
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document.write( "    ||u - v||^2 = (u,u) - 2(u,v) + (v,v)      (2)\r\n" );
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document.write( "Here (u,v) is the scalar product of vectors u and v, or, in your designations, (u,v) = u · v.\r\n" );
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document.write( "Subtract equation (2) from equation (1) .  You will get \r\n" );
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document.write( "    ||u + v||^2 − ||u − v||^2 = 4u · v.\r\n" );
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document.write( "It is exactly what should be proven in part (a).\r\n" );
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document.write( "             Part (b)\r\n" );
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document.write( "Vectors u and v are orthogonal if and only if  u · v = 0.\r\n" );
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document.write( "In part (a), we show that  4u · v = ||u + v||^2 − ||u − v||^2.\r\n" );
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document.write( "So, u · v = 0  if and only if  ||u + v||^2 − ||u − v||^2 = 0,  or  ||u + v||^2 = ||u − v||^2.\r\n" );
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document.write( "It is exactly what should be proven in part (b).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, proved and completed.\r
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