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Dot products of the vectors {\bar  {i}}+{\bar  {j}}-3{\bar  {k}},{\bar  {i}}+3{\bar  {j}}-2{\bar  {k}},2{\bar  {i}}+{\bar  {j}}+4{\bar  {k}}\, are 0,5,8\, respectively.Find the vector.

Let a,b,c be the given vectors and {\bar  {r}}=x{\bar  {i}}+y{\bar  {j}}+z{\bar  {k}}\, be the required vector.

Given {\bar  {r}}\cdot {\bar  {a}}=0,{\bar  {r}}\cdot {\bar  {b}}=5,{\bar  {r}}\cdot {\bar  {c}}=8\,

Now x+y-3z=0,x+3y-2z=5,2x+y+4z=8\,

Solving the first two equations and the second two,we get 2y+z=0,5y-8z=2\,

Solving these two equations,we get y=2,z=1\,

Solving for x,we get x=1.

Therefore,the required vector is {\bar  {i}}+2{\bar  {j}}+{\bar  {k}}\,

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