The variable r in a vector equation represents the magnitude or length of the vector, while $\hat{r}$ represents the direction of the vector. In other words, r tells us how long the vector is, while $\hat{r}$ tells us the direction in which it is pointing.
In a vector equation, r and $\hat{r}$ are related by the formula r$\hat{r}$, where r is the magnitude and $\hat{r}$ is the direction. This means that the vector can be written as a scalar (r) multiplied by a unit vector ($\hat{r}$) in the direction of the vector.
Yes, r and $\hat{r}$ can have different units in a vector equation. The magnitude (r) can have units such as meters or seconds, while the direction ($\hat{r}$) is dimensionless and does not have any units.
In a vector equation, r and $\hat{r}$ are important because they provide information about the magnitude and direction of the vector. They allow us to represent a vector in a concise and compact manner, making it easier to perform calculations and analyze the vector.
The relationship between r and $\hat{r}$ is useful in vector algebra because it allows us to perform operations on vectors in an efficient manner. For example, we can easily add or subtract vectors by combining the magnitudes (r values) and using the direction ($\hat{r}$) to determine the resulting vector. This relationship also allows us to convert between different coordinate systems and perform other vector operations.