Can You Apply Cross or Dot Product to Both Sides of a Vector Equation?

Mathematics often involves operations with vector equations. One particularly intriguing question arises when considering whether it is permissible to apply vector operations such as the cross product or dot product to both sides of a vector equation.

Understanding Vector Equations and Their Equality

Suppose we have a vector equation of the form fx gx for all x in some unknown set of vectors. Here, it is crucial to understand that the equality symbol signifies that the vector fx is identical to the vector gx for each x in the specified set. This identity guarantees that any vector operation applied to identical vectors will preserve this equality.

The Implications for Vector Operations

Given the equality fx gx, it follows that any vector operation, such as the dot product or cross product, will produce identical results on both sides of the equation. For instance, applying the dot product to both sides would yield:

[mathbf{fx} cdot mathbf{fx} mathbf{gx} cdot mathbf{gx}]

and similarly for the cross product:

[mathbf{fx} times mathbf{fx} mathbf{gx} times mathbf{gx}]

Example with the Dot Product

Consider a specific vector equation such as F(x) G(x). If we apply the dot product to both sides, we get:

[mathbf{F(x)} cdot mathbf{F(x)} mathbf{G(x)} cdot mathbf{G(x)}]

This means that the operation preserves the equality and the resulting identities remain valid.

Example with the Cross Product

In a similar vein, applying the cross product to both sides of the vector equation F(x) G(x) will yield:

[mathbf{F(x)} times mathbf{F(x)} mathbf{G(x)} times mathbf{G(x)}]

Once again, the properties of vector operations ensure that the resulting identities are preserved.

Reversibility of Identities

While applying vector operations to both sides of a vector equation preserves the existing equality, it does not automatically reverse or provide additional information about the original vectors. It is important to note that these operations yield identical vector-valued results and thus preserve the original equality without introducing new information or reversing the resulting identities.

Conclusion

In summary, you can indeed apply the cross product or dot product to both sides of a vector equation. These operations are well-defined and preserve the equality. However, the operation of applying a vector operation to both sides of a vector equation does not reverse or provide additional information beyond the original equality.

Additional Resources for Deeper Understanding

If you're interested in delving deeper into vector equations and operations, consider exploring the following resources:

Axler, S. (2015). Liner Algebra Done Right. Springer. Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.

These texts provide comprehensive insights into vector spaces, operations, and their applications.