The quantity \(\vec{E} \cdot \vec{B}\) is invariant. What does this say about how different observers will measure the angle between \(\vec{E}\) and \(\vec{B}\) in a light wave?

When we talk about the electric field vector, \( \vec{E} \) in physics, we're referring to a quantity that represents both the magnitude and the direction of the electric field at any point in space. Think of it as an arrow pointing from one place to another, where the length of the arrow represents how strong the electric field is, and the direction of the arrow shows the direction a positive charge would move if placed in that field.

For example, around a positive charge, the electric field vectors point away from the charge. The electric field is foundational in electromagnetism, and understanding its vector nature is crucial because it helps us calculate forces on charges and work done by the field when charges move from one point to another.

Similarly, a magnetic field vector, \( \vec{B} \), illustrates the direction and strength of a magnetic field at a particular point. Like the electric field vector, this is also depicted as an arrow; however, it represents different concepts. The direction of the magnetic field vector is the direction a north pole of a magnet would point if placed within the field.

In case of a simple bar magnet, these vectors would emerge from the magnet's north pole and enter at the south pole, forming a loop outside the magnet. Understanding the magnetic field vector is not only important in basic physics but also essential for complex applications such as electrical engineering and magnetic resonance imaging (MRI).

The vector dot product, also known as the scalar product, is a way to multiply two vectors that results in a scalar (a single number). It's calculated by multiplying the magnitudes of the two vectors and then taking the cosine of the angle between them. This is why it is sometimes said that the dot product measures how much one vector 'points' in the same direction as the other vector.

Mathematically, it's expressed as \( \vec{A} \cdot \vec{B} = |\vec{A}|\ |\vec{B}|\ \cos(\theta) \), where \( \vec{A} \) and \( \vec{B} \) are vectors, \( |\vec{A}| \) and \( |\vec{B}| \) are their magnitudes, and \( \theta \) is the angle between them. In the context of electromagnetic waves, if \( \vec{E} \) and \( \vec{B} \) are perpendicular, the dot product is zero because the cosine of 90 degrees is zero.

Invariance is a term in physics that denotes a property remains unchanged under specific transformations or observations. For instance, the invariance of the dot product \( \vec{E} \cdot \vec{B} \) in an electromagnetic wave implies that no matter where you are or how you're moving, the angle between the electric and magnetic field vectors will seem to remain the same—perpendicular in this case.

This principle of invariance is deeply integrated into the theories of relativity and conservation laws. It underwrites the fundamental symmetries of the universe and assures us that certain quantities are conserved across different frames of reference. For students, understanding invariance is key when studying the consistency of physical laws in various reference frames, like when considering observers moving at different velocities.