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Question
If `bar "a" = hat"i" + 2hat"j" + 3hat"k" , bar"b" = 3hat"i" + 2hat"j"` and `bar"c" = 2hat"i" + hat"j" + 3hat"k"`, then verify that `bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"`
Solution
`bar"b" xx bar"c" = |(hat"i",hat"j",hat"k"),(3,2,0),(2,1,3)|`
`= (6 - 0)hat"i" - (9 - 0)hat"j" + (3 - 4)hat"k"`
`= 6hat"i" - 9hat"j" - hat"k"`
∴ `bar"a" xx (bar"b" xx bar"c") = |(hat"i",hat"j", hat"k"),(1,2,3),(6,-9,-1)|`
`= (- 2 + 27)hat"i" - (- 1 - 18)hat"j" + (- 9 - 12)hat"k"`
`= 25hat"i" + 19hat"j" - 21hat"k"` ...(1)
`bar"a" . bar"c" = (hat"i" + 2hat"j" + 3hat"k").(2hat"i" + hat"j" + 3hat"k")`
= (1)(2) + (2)(1) + (3)(3)
= 2 + 2 + 9 = 13
∴ `(bar"a" . bar"c").bar"b" = 13(3hat"i" + 2hat"j") = 39hat"i" + 26hat"j"`
Also, `(bar"a" . bar"b") = (hat"i" + 2hat"j" + 3hat"k").(3hat"i" + 2hat"j")`
= (1)(3) + (2)(2) + (3)(0)
= 3 + 4 + 0 = 7
∴`(bar"a" . bar"b").bar"c" = 7(2hat"i" + hat"j" + 3hat"k") = 14hat"i" + 7hat"j" + 21hat"k"`
∴`(bar"a" . bar"c").bar"b" - (bar"a" . bar"b").bar"c"`
`= (39hat"i" + 26hat"j") - (14hat"i" + 7hat"j" + 21hat"k")`
`= 25hat"i" + 19hat"j" - 21hat"k"` .....(2)
From (1) and (2), we get
`bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"`
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