Which of the following expressions are meaningful?

Options

A. `vec(u).(vec(v)xxvec(w))`
B. `(vec(u).vec(v)).vec(w)`
C. `(vec(u).vec(v))vec(w)`
D. `vec(u)xx(vec(v).vec(w))`

Answer

To determine which of the given expressions are meaningful, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Option 1: \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \)**
- Here, \( \mathbf{v} \times \mathbf{w} \) is the cross product of vectors \( \mathbf{v} \) and \( \mathbf{w} \), resulting in a vector that is perpendicular to both \( \mathbf{v} \) and \( \mathbf{w} \).
- The expression then becomes \( \mathbf{u} \cdot \text{(a vector)} \), which is a valid dot product operation.
- **Conclusion:** This expression is meaningful.

2. **Option 2: \( \mathbf{u} \cdot \mathbf{v} \cdot \mathbf{w} \)**
- The first part \( \mathbf{u} \cdot \mathbf{v} \) results in a scalar (magnitude).
- The expression then attempts to take the dot product of this scalar with \( \mathbf{w} \), which is not valid since a scalar cannot be dotted with a vector.
- **Conclusion:** This expression is not meaningful.

3. **Option 3: \( (\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{w} \)**
- Similar to option 2, \( \mathbf{u} \cdot \mathbf{v} \) gives a scalar.
- However, the expression is interpreted as multiplying this scalar with the vector \( \mathbf{w} \), which is valid.
- For example, if \( \mathbf{u} \cdot \mathbf{v} = k \) (a scalar), then \( k \cdot \mathbf{w} \) is a meaningful operation.
- **Conclusion:** This expression is meaningful.

4. **Option 4: \( \mathbf{u} \times (\mathbf{v} \cdot \mathbf{w}) \)**
- Here, \( \mathbf{v} \cdot \mathbf{w} \) results in a scalar.
- The expression then tries to take the cross product of vector \( \mathbf{u} \) with this scalar, which is not valid since a vector cannot be crossed with a scalar.
- **Conclusion:** This expression is not meaningful.

### Final Answer:
The meaningful expressions are **Option 1** and **Option 3**.