LinAlg4.2.4

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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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