Question

If `bar"a" = hat"i" - 2hat"j" + 3hat"k"`  , `bar"b" = 4hat"i" - 3hat"j" + hat"k"` , `bar"c" = hat"i" - hat"j" + 2hat"k"` verify that `bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"`

Answer 1

Given: `bar"a" = hat"i" - 2hat"j" + 3hat"k"`  , `bar"b" = 4hat"i" - 3hat"j" + hat"k"` , `bar"c" = hat"i" - hat"j" + 2hat"k"`

∴ `bar"b" + bar"c" = (4hat"i" - 3hat"j" + hat"k") + (hat"i" - hat"j" + 2hat"k")`

`= 5hat"i" - 4hat"j" + 3hat"k"`

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

`= (- 6 + 12)hat"i" - (3 - 15)hat"j" + (- 4 +10)hat"k"`

`= 6hat"i" + 12hat"j" + 6hat"k"`    ...(1)

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

`= (- 2 + 9)hat"i" - (1 - 12)hat"j" + (- 3 + 8)hat"k"`

`= 7hat"i" + 11hat"j" + 5hat"k"`

and `bar"a" xx bar"c" = |(hat"i",hat"j",hat"k"),(1,-2,3),(1,-1,2)|`

`= (- 4+3)hat"i" - (2 - 3)hat"j" + (- 1 + 2)hat"k"`

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

∴ `bar"a" xx bar"b" + bar"a" xx bar"c" = (7hat"i" + 11hat"j" + 5hat"k") + (- hat"i" + hat"j" + hat"k")`

`= 6hat"i" + 12hat"j" + 6hat"k"`     .....(2)

From (1) and (2), we get

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