1. The problem involves vectors \(\mathbf{x}\) and \(\mathbf{y}\), which means we are dealing with vector operations rather than scalar algebra. 2. Common vector operations include addition, subtraction, dot product, and cross product. The choice depends on the context or the specific question. 3. For example, the dot product formula for vectors \(\mathbf{x} = (x_1, x_2, ..., x_n)\) and \(\mathbf{y} = (y_1, y_2, ..., y_n)\) is: $$\mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^n x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n$$ 4. The dot product results in a scalar and measures the extent to which two vectors point in the same direction. 5. If you want the cross product (only defined in 3D), for \(\mathbf{x} = (x_1, x_2, x_3)\) and \(\mathbf{y} = (y_1, y_2, y_3)\), the formula is: $$\mathbf{x} \times \mathbf{y} = (x_2 y_3 - x_3 y_2, x_3 y_1 - x_1 y_3, x_1 y_2 - x_2 y_1)$$ 6. The cross product results in a vector perpendicular to both \(\mathbf{x}\) and \(\mathbf{y}\). 7. Please specify the exact operation or problem involving vectors \(\mathbf{x}\) and \(\mathbf{y}\) for a detailed solution.