# 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.

#### 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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