The given vectors are $X_1 = (a,b,c); \ X_2=(b,c,a); \ X_3 = (c,a,b)$ and the resultant matrix will be.

$\begin{bmatrix} a &b &c \\b &c &a \\c &a &b \end{bmatrix}$

The vector are linearly dependent if the determinant of this matrix is zero,

i.e.

$\begin{vmatrix} a &b &c \\b &c &a \\c &a &b \end{vmatrix} = 0$

$\therefore a (bc - a^2) - b (b^2 - ac) + c (ab - c^2) =0$

$\therefore abc - a^3 - b^3 + abc +abc - c^3 = 0$

$\therefore a^3 + b^3 + c^3- 3abc = 0$

But we know that, if $a^3 + b^3 + c^3 - 3abc = 0$

Then, either $(a+b+c) =0$ or  $a=b=c$

Given that $(a+b+c) \ne 0$

Thus, we can conclude that the given vectors X1, X2, X3 are linearly dependent when a=b=c.

Otherwise they are linear independent.