To solve this problem, we need to understand the condition under which the linear combination of non-collinear vectors becomes zero. Given the equation:
\(x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}\),
where \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are non-collinear vectors, and \(x\), \(y\), \(z\) are scalars. Let's explore the implications:
For a vector equation \(x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}\) involving non-collinear vectors, the only solution that satisfies this equation under all conditions (in the vector space) is when the scalars multiply to zero. This is because non-collinear vectors are linearly independent, meaning they cannot be expressed as a linear combination of each other except by multiplication with zero.
The condition for non-collinear vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) means no scalar multiple of one vector plus a scalar multiple of another vector can add up to become the third vector, unless we're considering the trivial case where each vector is multiplied by zero.
Thus, for the equation to hold true, each scalar must be equal to zero: \(x = 0\), \(y = 0\), and \(z = 0\). Otherwise, if any scalar were non-zero, it would imply a dependency between the vectors, contradicting their non-collinearity.
Therefore, the correct answer is \(x = 0, y = 0, z = 0\).