Basis Vectors In Cylindrical Coordinates
This animation builds cylindrical coordinates from a 3D Cartesian setup, showing a point P, its position vector r, and the projection that defines ρ, φ, and z. A right triangle reveals why x = ρcosφ and y = ρsinφ. The local unit vectors ρ-hat, φ-hat, and z-hat are drawn as rotating, point-dependent directions, then expressed in i, j, k components. It closes by deriving the coordinate basis vectors e1, e2, e3 from partial derivatives of r, showing why e2 = ρφ-hat is scaled by arc length. Useful for students beginning curvilinear coordinates.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
Create a beginner-friendly educational physics diagram titled: “Problem 1(a) — Basis Vectors in Cylindrical Coordinates” Follow the mathematics of the homework, but teach it visually to a student who is completely new to curvilinear coordinates. Start with a 3D Cartesian xyz coordinate system. Show a point P somewhere above the xy-plane. Draw the position vector r from the origin to P. Project P vertically down onto the xy-plane. Introduce cylindrical coordinates: ρ = distance from the z-axis to the projection of P φ = angle around the z-axis z = height of P. Show the Cartesian relationships: x = ρ cosφ y = ρ sinφ z = z Use a right triangle in the xy-plane to visually explain why x and y are the projections of ρ. Then introduce the three cylindrical basis directions: ρ-hat φ-hat z-hat Draw: ρ-hat pointing directly away from the z-axis φ-hat tangent to the circular path around the z-axis z-hat vertically upward. Explain visually that these are LOCAL directions: they belong to the point P and can rotate as P moves. Then show the rectangular-basis expressions: ρ-hat = cosφ i-hat + sinφ j-hat φ-hat = −sinφ i-hat + cosφ j-hat z-hat = k-hat Do not just display the equations. Draw each vector and show its x and y projections. Finally introduce the actual coordinate basis vectors: e1 = ∂r/∂ρ = ρ-hat e2 = ∂r/∂φ = ρ φ-hat e3 = ∂r/∂z = z-hat Visually explain why e2 is longer than φ-hat: changing the angle φ moves the point along a circular arc, and the arc length depends on ρ. Main takeaway at the bottom: “A basis vector tells us what physical displacement happens when one coordinate changes.”
Make the next one in this series
- Show how spherical coordinates basis vectors r-hat, theta-hat, phi-hat are derived the same way
- Animate how velocity and acceleration look in cylindrical coordinates using e1, e2, e3
- Visualize the metric tensor components g_ij from the coordinate basis vectors in cylindrical coordinates
