There is nothing to explain. You misquoted the solution. It says [itex]\hat{r}[/itex]=r[itex]\hat{e}_{r}[/itex]. In plain English it says "Vector r is the same as scalar r times a unit vector in the direction of r." I am not sure why there is a "hat" over r instead of an arrow thus making the magnitude r =1, but it is what it is.athrun200 said:
Displacement vector is a type of vector that represents the change in position of an object from its initial position to its final position. It is different from other types of vectors, such as velocity or acceleration vectors, as it only considers the change in position and not the time or rate of change.
Displacement vector is represented by a bold letter, such as d, or by an arrow on top of the letter, such as →d. It can also be represented using its components, x and y, in the form of d = (x, y) or →d = (x, y).
A unit vector, denoted by &hat;r, is a vector with a magnitude of 1 and is used to indicate the direction of the displacement vector. It allows us to specify the direction of the displacement without affecting its magnitude, making it easier to understand and work with mathematically.
In this notation, r represents the magnitude of the displacement vector and &hat;eθ represents the unit vector in the direction of the displacement. To find the displacement vector, we simply multiply the magnitude r by the unit vector &hat;eθ. This gives us the displacement vector in the form of d = r&hat;eθ or →d = r&hat;eθ.
For example, let's say an object moves 5 meters in the direction of 30 degrees from its initial position. Using the &hat;r = r&hat;eθ notation, the displacement vector would be d = (5 m)&hat;e30°. This means that the magnitude of the displacement is 5 meters and it is directed at an angle of 30 degrees from the reference axis.
