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If vector a =i+j+k , b=4i+3j+4k and c=i+(alpha) j +(beta ) k are linearly dependent vectors and |c|=square root 3

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Solution:    Given that   veca=i+j+k , vecb=4i+3j+4k, hspace0.2cmvecc=i+alpha j+eta k

            are linearly dependent 

                                            eginvmatrix 1 & 1 &1 \ 4 &3 &4 \ 1 &alpha &eta endvmatrix=0 hspace0.5cmRightarrow 1-eta=0Rightarrow eta=1

        Also given that     left | vecc 
ight |=sqrt3Rightarrow 1+alpha^2+eta^2=3

     Substituting  the value of  eta , we get      Rightarrow hspace1cmalpha^2=1Rightarrow hspace0.2cmalpha=pm1

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Deependra Verma

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