Advertisement Remove all ads

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"`

Advertisement Remove all ads

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
  Is there an error in this question or solution?
Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×