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The correct answer is option **B**, \( k \mathbf{u} \times \mathbf{v} \).

Here's why:

- **Option A**: \( k^2 \mathbf{u} \cdot \mathbf{v} \) represents a scalar, not a vector, and hence cannot be orthogonal in the vector sense.
  
- **Option B**: \( k \mathbf{u} \times \mathbf{v} \) gives a vector that is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\) by the properties of the cross product. The scalar \( k \) simply scales this vector but does not change its direction.

- **Option C**: \( k \mathbf{u} \cdot \mathbf{v} \) is again a scalar and not a vector.

- **Option D**: \( k \mathbf{u} \mathbf{v} \) is not a valid vector operation as it suggests multiplication of two vectors without specifying the nature of the operation (dot, cross, etc.).

Would you like more details on how the cross product ensures orthogonality, or do you have any other questions? Here are a few additional questions you might consider:
1. What is the definition of the cross product?
2. How does the dot product differ from the cross product in terms of vector operations?
3. What would be the properties of a vector if we multiply it by a scalar?
4. Could a scalar multiple of a dot product between two vectors ever result in a vector?
5. What are some physical applications of the cross product?

Tip: Always remember that the cross product of two vectors results in a vector orthogonal to both, which is useful in determining directions perpendicular to planes defined by vectors.

If k is any constant, which of the following describes a vector orthogonal to u and v? A) k^2 * u · v B) k |u x v| C) k u · v D) k u x v

This problem covers vector orthogonality and how the cross product of two vectors results in a vector orthogonal to both. Learn the significance of the cross product in vector algebra and why option D (k u x v) is the correct answer.

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