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If the vectors a=hat(i)+a hat(j)+a^(2) hat(k), b=hat(i)+b hat(j)+b^(2) hat(k) and c=hat(i)+chat(j)+c^(2) hat(k) are three non-coplanar vectors and |(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),1+c^(3))|=0, then the value of abc is

Answer»

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`-1`

Solution :Since, a, B and C are non-coplanar vectoprs THEREFORE `[abc]NE0`
`RARR""Delta=|{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|ne0rArrDeltane0`
`"Now,"|{:(1,a^2,1+a^3),(1,b,1+b^3),(1,c^2,1+c^3):}|=0`
`rArr""|{:(1,a^2,1),(1,b,1),(1,c^2,1):}|+|{:(a,a^2,a^3),(b,b^2, b^3),(c,c^2,c^3):}|=0`
`rArr""Delta(1+abc)=0rArr""abc=-1[because Delta=ne]`


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