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If aijkbija¯=i^+2j^+3k^,b¯=3i^+2j^ and cijkc¯=2i^+j^+3k^, then verify that abcacbabca¯×(b¯×c¯)=(a¯.c¯)b¯-(a¯.b¯)c¯ - Mathematics and Statistics

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Sum

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"`

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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"`

Concept: Vector Triple Product
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