Make yourself a drawing with ##\hat{e}_x## horizontal and ##\hat{e}_y## vertical. In this drawing add ##\hat{e}_r## at angle ##\theta## w.r.t. ##\hat{e}_x##. Add ##\hat{e}_{\theta}## perpendicular to ##\hat{e}_r##. Study the drawing and find the projections of ##\hat{e}_x## on ##\hat{e}_r## and ##\hat{e}_{\theta}##. You should get what you expect to get.Komekami said:How do I express ex,ey,ez in terms er,eθ,eZ?
Cartesian unit vectors are three orthogonal vectors (i, j, k) that represent the three dimensions of a Cartesian coordinate system. These vectors have a magnitude of 1 and are used to describe the direction and orientation of a point or vector in space.
Cylindrical vectors are a set of three vectors (ρ, φ, z) used in cylindrical coordinate systems to describe the position and orientation of a point or vector in space. The ρ vector represents the distance from the origin, the φ vector represents the angle from a reference plane, and the z vector represents the height or depth.
Cartesian unit vectors can be converted into cylindrical vectors using trigonometric functions. The ρ vector is equivalent to the x and y components of the Cartesian vector, while the φ vector is equivalent to the angle between the Cartesian x-y plane and the cylindrical ρ vector. The z vector remains the same in both coordinate systems.
The magnitudes of Cartesian unit vectors and cylindrical vectors are equal. This means that the length of the Cartesian unit vector (i, j, k) is equal to the length of the cylindrical vector (ρ, φ, z). Both have a magnitude of 1.
Yes, cylindrical vectors can be converted back to Cartesian unit vectors using the same trigonometric functions. The x and y components can be found using the ρ and φ vectors, and the z component remains the same. The resulting Cartesian unit vector will have a magnitude of 1 and represent the same direction and orientation in space.