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Question
If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = 3hat"i" + 5hat"j" + 2hat"k", vec"c" = - hat"i" - 2hat"j" + 3hat"k"`, verify that `(vec"a" xx vec"b") xx vec"c" = (vec"a"*vec"c")vec"b" - (vec"b" * vec"c")vec"a"`
Solution
L.H.S = `vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(2, 3, -1),(3"m", 5, 2)|`
= `hat"i"(6 + 5) - hat"j"(4 + 3) + hat"k"(10 - 9)`
= `11hat"i" - 7hat"j" + hat"k"`
`(vec"a" xx vec"b") xx vec"c" = |(hat"i", hat"j", hat"k"),(11, -7, 1),(-1, -2, 3)|`
= `hat"i"(-21 + 2) - hat"j"(33 + 1) + hat"k"(- 22 - 7)`
= `-19hat"i" - 34hat"j" - 29hat"k"` .........(1)
R.H.S
`vec"a" * vec"c" = (2hat"i" + 3hat"j" - hat"k")*(-hat"i" - 2hat"j" + 3hat"k")`
= 2 – 6 – 3
= – 11
`(vec"a" * vec"c")vec"b" = -11(3hat"i" + 5hat"j" + 2hat"k")`
= `-33hat"i" - 55hat"j" - 22hat"k"`
`vec"b" * vec"c" = (3hat"i" + 5hat"j" + 2hat"k")*(-1hat"i" - 2hat"j" + 3hat"k")`
= – 3 – 10 + 6
= – 7
`(vec"b" * vec"c")vec"a" = -7(2hat"i" + 3hat"j" - hat"k")`
= `-14hat"i" - 21hat"j" + 7hat"k"`
`(vec"a" * vec"c")vec"b" - (vec"b" * vec"c")vec"a" = - 33hat"i" - 55hat"j" - 22hat"k" + 14hat"i" + 21hat"j" - 7hat"k"`
= `- 19hat"i" - 34hat"j" - 29hat"k"` ..........(2)
By (1) and (2)
`(vec"a" xx vec"b") xx vec"c" = (vec"a"*vec"c")vec"b" - (vec"b" * vec"c")vec"a"`
`vec"a" xx (vec"b" xx vec"c") = (vec"a"*vec"c")vec"b" - (vec"a"*vec"b")vec"c"`
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