Prove that a × [a × (a × b)] = (a.a)(b × a).
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QUESTION
Prove that a × [a × (a × b)] = (a.a)(b × a).
ANSWER
FORMULA USED
A×(B×C)=(A.C)B - (A.B)C
SOLUTION.
a × [a × (a × b)]
= a×[(a.b)a-(a.a)b]
=[ax(a.b)a-(a.a)(a×b]
Now, a×(a.b)=0
because vectors are coplanar
=> [0-(a.a)(a×b)]
=-(a.a)(a×b)
Now , using property
(a×b)=-(b×a)
=>-(a.a)(a×b)
=(a.a)(-)(-)(b×a)
= (a.a)(b × a).
HENCE PROVED
Prove that a × [a × (a × b)] = (a.a)(b × a).
ANSWER
FORMULA USED
A×(B×C)=(A.C)B - (A.B)C
SOLUTION.
a × [a × (a × b)]
= a×[(a.b)a-(a.a)b]
=[ax(a.b)a-(a.a)(a×b]
Now, a×(a.b)=0
because vectors are coplanar
=> [0-(a.a)(a×b)]
=-(a.a)(a×b)
Now , using property
(a×b)=-(b×a)
=>-(a.a)(a×b)
=(a.a)(-)(-)(b×a)
= (a.a)(b × a).
HENCE PROVED
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