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Given-vector-a-i-2j-k-b-2i-j-2k-c-i-3j-k-and-d-2j-2k-Find-the-value-of-a-b-c-d-

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Question Number 133925 by bemath last updated on 25/Feb/21
Given vector a^→ = i^� −2j^� +k^� , b^→ = 2i^� +j^� −2k^� , c^→ =−i^� +3j^� −k^� and d^→ = 2j^� −2k^� . Find the value of (a^→ ×b^→ )×(c^→ ×d^→ ).
Givenvectora→=i^−2j^+k^,b→=2i^+j^−2k^,c→=−i^+3j^−k^andd→=2j^−2k^.Findthevalueof(a→×b→)×(c→×d→).
Answered by EDWIN88 last updated on 25/Feb/21
by use identity : (a×b)×(c×d)= b(acd)−a(bcd) where acd = a(c×d) and bcd =b(c×d) (1) c×d = determinant (((−1 3 −1)),(( 0 2 −2)))= −4i−2j−2k then acd = (1,−2,1).(−4,−2,−2)=−4+4−2=−2 bcd = (2,1,−2).(−4,−2,−2)=−8−2+4=−6 then we get (a×b)×(c×d)=−2b+6a = ((( 6)),((−12)),(( 6)) ) − ((( 4)),(( 2)),((−4)) ) = 2i−14j+10k
byuseidentity:(a×b)×(c×d)=b(acd)−a(bcd)whereacd=a(c×d)andbcd=b(c×d)(1)c×d=|−13−102−2|=−4i−2j−2kthenacd=(1,−2,1).(−4,−2,−2)=−4+4−2=−2bcd=(2,1,−2).(−4,−2,−2)=−8−2+4=−6thenweget(a×b)×(c×d)=−2b+6a=(6−126)−(42−4)=2i−14j+10k
Answered by bemath last updated on 25/Feb/21
(1) a×b = determinant (((1 −2 1)),((2 1 −2)))= 3i+4j+5k (2) c×d = determinant (((−1 3 −1)),(( 0 2 −2)))= −4i−2j−2k then (a×b)×(c×d)= determinant ((( 3 4 5)),((−4 −2 −2))) = 2i−14j+10k
(1)a×b=|1−2121−2|=3i+4j+5k(2)c×d=|−13−102−2|=−4i−2j−2kthen(a×b)×(c×d)=|345−4−2−2|=2i−14j+10k

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