1. The problem is to understand what is wrong with the expression $\mathbf{u} \times \mathbf{v} \times \mathbf{w}$ where $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ are vectors. 2. In vector algebra, the multiplication of vectors can be of two main types: the dot product (scalar product) and the cross product (vector product). 3. The dot product $\mathbf{a} \cdot \mathbf{b}$ results in a scalar, while the cross product $\mathbf{a} \times \mathbf{b}$ results in another vector. 4. The expression $\mathbf{u} \times \mathbf{v} \times \mathbf{w}$ is ambiguous because the cross product is a binary operation and is not associative. This means: $$\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) \neq (\mathbf{u} \times \mathbf{v}) \times \mathbf{w}$$ 5. Therefore, writing $\mathbf{u} \times \mathbf{v} \times \mathbf{w}$ without parentheses is incorrect and unclear. 6. To correctly evaluate such an expression, you must specify the order of operations using parentheses. 7. For example, the vector triple product identity states: $$\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = \mathbf{v} (\mathbf{u} \cdot \mathbf{w}) - \mathbf{w} (\mathbf{u} \cdot \mathbf{v})$$ 8. This identity is useful and shows that the triple cross product can be expressed in terms of dot products and scalar multiplication. 9. In summary, the problem with the expression is the lack of parentheses and the misunderstanding that the cross product is not associative. Final answer: The expression $\mathbf{u} \times \mathbf{v} \times \mathbf{w}$ is invalid without parentheses because the cross product is not associative. You must write either $\mathbf{u} \times (\mathbf{v} \times \mathbf{w})$ or $(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}$ to be mathematically correct.