# 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

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"

#### 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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