= Solution
For $\mathbf b=0$, a nonzero coefficient <vector> $(x,y,z)$ with $x\mathbf c_1+y\mathbf c_2+z\mathbf c_3=0$ is precisely a linear dependence of the three columns. Three <vectors> in three dimensions are linearly dependent exactly when the <determinant> of their column <matrix>, or their <scalar triple product>, is zero. Thus
$$
\boxed{\text{a nontrivial solution requires }[\mathbf c_1,\mathbf c_2,\mathbf c_3]=0.}
$$
For this square homogeneous <linear system>, the condition is also sufficient.
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