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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"a"*vec"b")vec"c"`
Solution
L.H.S
`vec"b" xx vec"c" = |(hat"i", vec"j", vec"k"),(3, 5, 2),(-1, -2, 3)|`
= `hat"i"(15 + 4) - hat"j"(9 + 2) + hat"k"(-6 + 5)`
= `19hat"i" - 11hat"j" - hat"k"`
`vec"a" xx (vec"b" xx vec"c") = |(hat"i", hat"j", hat"k"),(2, 3, -1),(19, -11, -1)|`
= `hat"i"(- 3 - 11) - hat"j"(- 2 + 19) + hat"k"(- 22 - 57)`
= `-14hat"i" - 17hat"j" - 79hat"k"` ........(1)
R.H.S
`vec"a"*vec"c"` = – 2 – 6 – 3 = – 11
`(vec"a"*vec"c")vec"b" = -11(3hat"i" + 5hat"j" + 2hat"k")`
= `-33hat"i" - 55hat"j" - 22hat"k"`
`vec"a"*vec"b"` = 6 + 15 – 2 = 19
`(vec"a"*vec"b")vec"c" = 19(- hat"i" - 2hat"j" + 3hat"k")`
= `- 19hat"i" - 38hat"j" + 57hat"k"`
`(vec"a"*vec"c")vec"b" - (vec"a"*vec"b")vec"c" = -33hat"i" - 55hat"j" - 22hat"k" + 19hat"i" + 38hat"j" - 57hat"k"`
= `14hat"i" - 17hat"j" - 79hat"k"` ........(2)
By (1) and (2)
`vec"a" xx (vec"b" xx vec"c") = (vec"a"*vec"c")vec"b" - (vec"a"*vec"c")vec"b"`
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