document.write( "Question 1206236: Prove that two vectors must have equal magnitudes if their sum is perpendicular to their difference \n" ); document.write( "
Algebra.Com's Answer #843526 by math_tutor2020(3816)\"\" \"About 
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\n" ); document.write( "I'll use a period to indicate dot product.
\n" ); document.write( "A . B = A dot product B
\n" ); document.write( "where A and B are vectors.\r
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\n" ); document.write( "\n" ); document.write( "Here are the dot product rules to memorize
  • A.B = B.A
  • A.(B+C) = A.B+A.C
  • 0.A = 0
  • |A| = sqrt(A.A)
  • (k*A).B = k*(A.B) = A.(k*B) where k is a scalar and * means the usual multiplication
Let
\n" ); document.write( "S = A+B
\n" ); document.write( "represent the sum of the vectors\r
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\n" ); document.write( "\n" ); document.write( "Recall that vectors are perpendicular if and only if their dot product is zero.
\n" ); document.write( "We'll start with the idea A+B is perpendicular to A-B.
\n" ); document.write( "That must mean (A + B) . (A - B) = 0
\n" ); document.write( "The goal is to end up with |A| = |B| to show they have the same magnitude.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "(A + B) . (A - B) = 0
\n" ); document.write( "S . (A - B) = 0
\n" ); document.write( "S . A - S . B = 0
\n" ); document.write( "S . A = S . B
\n" ); document.write( "(A+B) . A = (A+B) . B
\n" ); document.write( "A . (A+B) = B . (A+B)
\n" ); document.write( "A.A + A.B = B.A + B.B
\n" ); document.write( "A.A + A.B = B.B + A.B
\n" ); document.write( "A.A = B.B
\n" ); document.write( "sqrt(A.A) = sqrt(B.B)
\n" ); document.write( "|A| = |B|\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We have shown that (A + B) . (A - B) = 0 leads to |A| = |B|
\n" ); document.write( "Therefore, if A+B is perpendicular to A-B, then vectors A and B must have the same magnitude.
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