Cylindrical and spherical coordinates each carry three unit vectors. Cylindrical keeps the polar pair and adds a fixed vertical direction: r̂, θ̂, and ẑ. Spherical replaces the whole set with r̂, θ̂, and φ̂, all three of which turn as you move. This is a reference post, so the tables below are the point, and the calculator is there whenever you want to confirm a set squares to 1.
Both systems extend the polar unit vectors into three dimensions. The difference is how they add the third direction: cylindrical stacks a fixed axis on top, spherical wraps everything around a center.
That single choice is what separates them in practice. A fixed axis keeps one direction constant no matter where you stand, so any motion parallel to it leaves two of your three unit vectors alone. Wrapping around a center removes that anchor: there is no direction that stays put, and every step you take reorients all three. Keep that split in mind as you read the tables, because it explains every sign and zero that follows.
Cylindrical: the polar pair plus a fixed axis
Cylindrical coordinates (r, θ, z) describe a point by its distance from a central axis, its angle around that axis, and its height. The first two unit vectors are exactly the polar ones, and the third is the ordinary vertical ẑ that never changes.
| Unit vector | Cartesian components | Points |
|---|---|---|
| r̂ | ⟨cos θ, sin θ, 0⟩ | Outward from the axis |
| θ̂ | ⟨−sin θ, cos θ, 0⟩ | Around the axis |
| ẑ | ⟨0, 0, 1⟩ | Along the axis |
Only r̂ and θ̂ rotate as you move around the axis. The ẑ stays put, which is what makes cylindrical coordinates comfortable: two-thirds of the work is just polar coordinates you already know.
The same length check applies here. At θ = 90° the set evaluates to r̂ = ⟨0, 1, 0⟩, θ̂ = ⟨−1, 0, 0⟩, and ẑ = ⟨0, 0, 1⟩, and each squares to 1.
All three pairwise dot products are 0 here as well, so the set is orthonormal, and at θ = 0 it collapses to exactly i, j, k.
Spherical: three directions that all move
Spherical coordinates describe a point by its distance from the origin and two angles. Here every unit vector depends on the angles, so all three turn as the point moves. This is the set worth memorizing carefully, because the sign pattern is easy to scramble.
| Unit vector | Cartesian components | Points |
|---|---|---|
| r̂ | ⟨sin θ cos φ, sin θ sin φ, cos θ⟩ | Outward from the origin |
| θ̂ | ⟨cos θ cos φ, cos θ sin φ, −sin θ⟩ | Toward increasing polar angle |
| φ̂ | ⟨−sin φ, cos φ, 0⟩ | Around the axis, increasing azimuth |
The table above uses the physics convention: θ is the polar angle measured down from the z-axis, and φ is the azimuthal angle in the xy-plane. Notice that φ̂ has no z-component at all, because moving around the axis never changes your height.
Read the columns as directions of motion, not just numbers. Increasing r pushes you straight out along r̂. Increasing θ tips you toward the equator along θ̂, which is why its z-component is negative in the upper hemisphere. Increasing φ swings you around the axis along φ̂, at constant height and constant distance from the origin. Each unit vector is the direction one coordinate grows while the other two hold still, and that reading survives the switch to any point on the sphere.
Verifying the spherical set
Pick a concrete point and check. At the polar angle θ = 45° with azimuth φ = 0, the three vectors come out to r̂ = ⟨0.7071, 0, 0.7071⟩, θ̂ = ⟨0.7071, 0, −0.7071⟩, and φ̂ = ⟨0, 1, 0⟩.
Test any one of these, say ⟨0.7071, 0, 0.7071⟩. Normalize it and confirm the length is 1.
Three lengths of 1. Now the perpendicularity, which every pair must satisfy for the set to be a valid basis.
| Pair | Dot product | Result |
|---|---|---|
| r̂ · θ̂ | (0.7071)(0.7071) + 0 + (0.7071)(−0.7071) | 0 |
| r̂ · φ̂ | (0.7071)(0) + (0)(1) + (0.7071)(0) | 0 |
| θ̂ · φ̂ | (0.7071)(0) + (0)(1) + (−0.7071)(0) | 0 |
All three dot products are 0, so the set is orthonormal at that point. The same holds at every point; the algebra reduces to sin²θ + cos²θ = 1 each time.
A second spherical point
One point can hide a coincidence, so move to a second one. Take the polar angle θ = 90° with azimuth φ = 90°, a point on the equator sitting above the positive y-axis.
| Vector | Components | Length |
|---|---|---|
| r̂ | ⟨0, 1, 0⟩ | 1 |
| θ̂ | ⟨0, 0, −1⟩ | 1 |
| φ̂ | ⟨−1, 0, 0⟩ | 1 |
All three lengths are 1, and the three pairwise dot products are 0, so the set stays orthonormal. The directions are easy to read off: r̂ points along +y, straight out to the point, θ̂ points straight down along −z toward the south pole, which is the direction of increasing polar angle, and φ̂ points along −x, around the axis.
Compare this with the first point, where φ̂ came out to ⟨0, 1, 0⟩ and pointed along +y. Same unit vector, different point, different direction in space: that is the whole character of the spherical set. Two points is enough to see that the orthonormality is not luck at a convenient angle. It holds because the length and dot-product algebra reduces to sin²θ + cos²θ = 1 and to matching sign-cancellations that never depend on the particular values you plug in.
From spherical coordinates back to a point
The unit vectors also give you the point itself. A point at radius r, polar angle θ, and azimuth φ sits at r times r̂, which in Cartesian components is (r sin θ cos φ, r sin θ sin φ, r cos θ).
Put in r = 2, θ = 60°, and φ = 30° and the point comes out to (1.5, 0.8660, 1). These are the same three expressions that make up r̂, scaled by r, which is why r̂ is called the radial direction: it is the direction the point lies in, seen from the origin.
The other two unit vectors do not enter the position at all, and that is the point of them. They describe how the position would move if you nudged an angle, not where it sits. So a full description of a point uses one radial length along r̂, while θ̂ and φ̂ stay in reserve for velocities, forces, and fields that push sideways rather than straight out. Reading the position formula this way keeps the roles of the three directions separate in your head.
The convention trap
This is the part that quietly breaks copied formulas. Physics and mathematics label the two spherical angles in opposite ways.
In the physics convention, used above and in the ISO standard, θ is the polar angle from the z-axis and φ is the azimuth. In the common mathematics convention, the two are swapped: θ is the azimuth and φ is the polar angle.
The vectors themselves are identical; only the letters change. But if you lift the component formulas from a physics source and read them with the math meaning of θ, every expression is wrong in a way no length check will catch, because a swapped-but-consistent set still has length 1. Fix the convention before you touch a formula, and confirm which one your own course uses. The Wikipedia article on the spherical coordinate system states both conventions side by side.
One habit removes most of the risk. When you copy a formula, write down next to it which angle is the polar one, then read your own diagram against that note before substituting anything. The letters can lie, but the geometry cannot: the polar angle is always the one measured from the axis, and the azimuth is always the one measured around it. Anchor to that picture and the label a given text happens to use stops mattering.
Where the systems show up
Two concrete cases make the split clear. The magnetic field around a long straight current-carrying wire circles the wire at every point, so it is naturally written along θ̂ in cylindrical coordinates, with the wire as the axis.
The electric field of a point charge is the spherical case: it points straight out along r̂ and weakens with distance. A planet’s gravity behaves the same way, pointing back along r̂ toward the center. Once you spot which direction the physics runs in, the coordinate choice follows.
The gain is that the field ends up with one nonzero component instead of three. Written in Cartesian coordinates, the wire’s field mixes x and y in a way that changes at every point, and the point charge’s field mixes all three. Written along θ̂ or r̂, each is a single term that depends only on distance. Matching the coordinates to the symmetry is what turns a messy expression into a short one, and it is the reason these unit vectors are worth carrying around.
Choosing a system
The geometry of the problem picks the coordinates. Cylindrical fits anything organized around a straight axis: the field around a current-carrying wire, flow through a pipe, stress in a rotating shaft. The fixed ẑ matches the fixed axis.
Spherical fits anything that radiates from a point: a planet’s gravity, the electric field of a point charge, sound spreading from a small source. There the natural directions are outward and around, which is exactly what r̂, θ̂, and φ̂ give you. If either set feels shaky, the two-dimensional polar unit vectors are the foundation both are built on, and the unit normal hub covers the fixed-direction case for contrast.
Confirm your course's convention before you trust a single formula. Once θ and φ are pinned down, the tables above are all you need.
Frequently Asked Questions
What are the spherical unit vectors?
Using the physics convention with θ as the polar angle and φ as the azimuth, they are r̂ = ⟨sin θ cos φ, sin θ sin φ, cos θ⟩, θ̂ = ⟨cos θ cos φ, cos θ sin φ, −sin θ⟩, and φ̂ = ⟨−sin φ, cos φ, 0⟩. All three have length 1 and are mutually perpendicular.
What are the cylindrical unit vectors?
They are the two polar unit vectors plus a fixed vertical one: r̂ = ⟨cos θ, sin θ, 0⟩, θ̂ = ⟨−sin θ, cos θ, 0⟩, and ẑ = ⟨0, 0, 1⟩.
Why do physics and math swap θ and φ?
The two fields adopted opposite labels. In physics θ is usually the polar angle from the z-axis and φ the azimuth; in much of mathematics the roles are reversed. A formula copied across conventions will be silently wrong, so check which one your source uses.
Which spherical unit vectors change with position?
All three. Unlike cylindrical coordinates, which keep a fixed ẑ, every spherical unit vector depends on the angles and points in a different direction as you move.
When should I use cylindrical versus spherical coordinates?
Use cylindrical when the problem has a straight axis, like a pipe, a wire, or a rotating shaft. Use spherical when it radiates from a central point, like a planet's gravity or a point charge's field.