Maharashtra State BoardHSC Arts 12th Board Exam
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If aija¯=i^-2j^, bijcijkb¯=i^+2j^,c¯=2i^+j^-2k^, then find (i) abca¯×(b¯×c¯) (ii) abc(a¯×b¯)×c¯ Are the results same? Justify. - Mathematics and Statistics

Sum

If `bar"a" = hat"i" - 2hat"j"`, `bar"b" = hat"i" + 2hat"j" , bar"c" = 2hat"i" + hat"j" - 2hat"k"`, then find (i) `bar"a" xx (bar"b" xx bar"c")` (ii) `(bar"a" xx bar"b") xx bar"c"` Are the results same? Justify.

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Solution

`bar"a" xx (bar"b" xx bar"c")`

`bar"b" xx bar"c" = |(hat"i",hat"j",hat"k"),(1,2,0),(2,1,-2)|`

`= (- 4 - 0)hat"i" - (- 2 - 0)hat"j" + (1 - 4)hat"k"`

`= - 4hat"i" + 2hat"j" - 3hat"k"`

∴ `bar"a" xx (bar"b" xx bar"c") = |(hat"i",hat"j",hat"k"),(1,-2,0),(- 4, 2, -3)|`

`= (6 - 0)hat"i" - ( - 3 - 0)hat"j" + (2 - 8)hat"k"`

`= 6hat"i" + 3hat"j" - 6hat"k"`

`(bar"a" xx bar"b") xx bar"c"`

`bar"a" xx bar"b" = |(hat"i",hat"j",hat"k"),(1,- 2,0),(1,2,0)|`

`= (0 - 0)hat"i" - (0 - 0)hat"j" + (2 - (- 2))hat"k"`

`= 4hat"k"`

∴ `(bar"a" xx bar"b") xx bar"c" = |(hat"i",hat"j",hat"k"),(0,0,4),(2, 1, -2)|`

`= (0 - 4)hat"i" - (0 - 8)hat"j" + (0 - 0)hat"k"`

`= - 4hat"i" + 8hat"j"`

`bar"a" xx (bar"b" xx bar"c") ≠ (bar"a" xx bar"b") xx bar"c"`

Concept: Vector Triple Product
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