If v_{1}=(1,1,1) ; v_{2}=(0,1,1) ; v_{3}=(0, x, 1) are linearly dependent vectors in R^{3} find the value for x_{6} ?

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Step:1 ：Form the matrix with the given vectors as columns and find its determinant: begin{vmatrix} 1 & 0 & 0  1 & 1 & x  1 & 1 & 1 end{vmatrix}Step:2 ：Calculate the determinant: (1)(1-1) - (0)(1-x) + (0)(x-1) = 0Step:3 ：(boxed{x = 1})

Answer

Step: 1 ：Form the matrix with the given vectors as columns and find its determinant: begin{vmatrix} 1 & 0 & 0  1 & 1 & x  1 & 1 & 1 end{vmatrix}Step: 2 ：Calculate the determinant: (1)(1-1) - (0)(1-x) + (0)(x-1) = 0Step: 3 ：(boxed{x = 1})

If $v_{1}=(1,1,1) ; v_{2}=(0,1,1) ; v_{3}=(0, x, 1)$ are linearly dependent vectors in $R^{3}$ find the value for $x_{6} ?$

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