Answer on Question #43543-Math-Vector Calculus
Prove that
( a ⃗ a 2 − b ⃗ b 2 ) 2 = ( a ⃗ − b ⃗ a b ) 2 . \left(\frac {\vec {a}}{a ^ {2}} - \frac {\vec {b}}{b ^ {2}}\right) ^ {2} = \left(\frac {\vec {a} - \vec {b}}{a b}\right) ^ {2}. ( a 2 a − b 2 b ) 2 = ( ab a − b ) 2 .
Solution
1.
( a ⃗ a 2 − b ⃗ b 2 ) 2 = ( a ⃗ a 2 ) 2 + ( b ⃗ b 2 ) 2 − 2 ⋅ a ⃗ a 2 ⋅ b ⃗ b 2 = a ⃗ 2 a 4 + b ⃗ 2 b 4 − 2 ( a ⃗ , b ⃗ ) a 2 b 2 = a 2 a 4 + b 2 b 4 − 2 ( a ⃗ , b ⃗ ) a 2 b 2 = 1 a 2 + 1 b 2 − 2 ( a ⃗ , b ⃗ ) a 2 b 2 . \begin{array}{l} \left(\frac {\vec {a}}{a ^ {2}} - \frac {\vec {b}}{b ^ {2}}\right) ^ {2} = \left(\frac {\vec {a}}{a ^ {2}}\right) ^ {2} + \left(\frac {\vec {b}}{b ^ {2}}\right) ^ {2} - 2 \cdot \frac {\vec {a}}{a ^ {2}} \cdot \frac {\vec {b}}{b ^ {2}} = \frac {\vec {a} ^ {2}}{a ^ {4}} + \frac {\vec {b} ^ {2}}{b ^ {4}} - \frac {2 (\vec {a} , \vec {b})}{a ^ {2} b ^ {2}} = \frac {a ^ {2}}{a ^ {4}} + \frac {b ^ {2}}{b ^ {4}} - \frac {2 (\vec {a} , \vec {b})}{a ^ {2} b ^ {2}} \\ = \frac {1}{a ^ {2}} + \frac {1}{b ^ {2}} - \frac {2 (\vec {a} , \vec {b})}{a ^ {2} b ^ {2}}. \\ \end{array} ( a 2 a − b 2 b ) 2 = ( a 2 a ) 2 + ( b 2 b ) 2 − 2 ⋅ a 2 a ⋅ b 2 b = a 4 a 2 + b 4 b 2 − a 2 b 2 2 ( a , b ) = a 4 a 2 + b 4 b 2 − a 2 b 2 2 ( a , b ) = a 2 1 + b 2 1 − a 2 b 2 2 ( a , b ) .
2.
( a ⃗ − b ⃗ a b ) 2 = ( a ⃗ − b ⃗ ) 2 a 2 b 2 = a ⃗ 2 + b ⃗ 2 − 2 ⋅ a ⃗ ⋅ b ⃗ a 2 b 2 = a 2 + b 2 − 2 ( a ⃗ , b ⃗ ) a 2 b 2 = 1 a 2 + 1 b 2 − 2 ( a ⃗ , b ⃗ ) a 2 b 2 . \left(\frac {\vec {a} - \vec {b}}{a b}\right) ^ {2} = \frac {\left(\vec {a} - \vec {b}\right) ^ {2}}{a ^ {2} b ^ {2}} = \frac {\vec {a} ^ {2} + \vec {b} ^ {2} - 2 \cdot \vec {a} \cdot \vec {b}}{a ^ {2} b ^ {2}} = \frac {a ^ {2} + b ^ {2} - 2 (\vec {a} , \vec {b})}{a ^ {2} b ^ {2}} = \frac {1}{a ^ {2}} + \frac {1}{b ^ {2}} - \frac {2 (\vec {a} , \vec {b})}{a ^ {2} b ^ {2}}. ( ab a − b ) 2 = a 2 b 2 ( a − b ) 2 = a 2 b 2 a 2 + b 2 − 2 ⋅ a ⋅ b = a 2 b 2 a 2 + b 2 − 2 ( a , b ) = a 2 1 + b 2 1 − a 2 b 2 2 ( a , b ) .
That's why
( a ⃗ a 2 − b ⃗ b 2 ) 2 = ( a ⃗ − b ⃗ a b ) 2 . \left(\frac {\vec {a}}{a ^ {2}} - \frac {\vec {b}}{b ^ {2}}\right) ^ {2} = \left(\frac {\vec {a} - \vec {b}}{a b}\right) ^ {2}. ( a 2 a − b 2 b ) 2 = ( ab a − b ) 2 .
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#339914 on Dec 2023