Enter vector details
Use any consistent unit for x, y, and z. The radius uses that same unit.
Formula used
The calculator treats your input as the Cartesian vector (x, y, z). It uses the scientific spherical convention below.
θ is measured down from the positive z-axis. φ rotates from positive x toward positive y. The atan2 function keeps the correct quadrant.
How to use this calculator
- Enter the x, y, and z components of one vector.
- Select degrees or radians for the angular output.
- Choose a positive or signed azimuth range.
- Set the number of decimal places you need.
- Select Convert Vector to show results above the form.
- Download a CSV file or save the result as PDF.
Example data table
| Vector (x, y, z) | Radius r | Inclination θ | Azimuth φ |
|---|---|---|---|
| (3, 4, 0) | 5 | 90° | 53.130° |
| (0, 0, 5) | 5 | 0° | 0° by convention |
| (1, -1, -1) | 1.732 | 125.264° | 315° positive range |
| (-2, 0, 2) | 2.828 | 45° | 180° |
Understanding vector conversions
Why spherical form helps
Vector to spherical conversion changes a three-dimensional Cartesian vector into radius and two angles. Cartesian form uses x, y, and z components. Spherical form describes length and direction. It is useful when direction matters more than separate components. Engineers use it for forces, fields, motion, and antenna patterns. Students use it in geometry, physics, graphics, and calculus. The conversion keeps the same vector. Only its description changes.
Radius and direction
The radius is the vector magnitude. It measures the straight distance from the origin. A larger radius means a longer vector. The inclination angle shows vertical direction. It is measured from the positive z-axis. The azimuth angle shows horizontal direction. It begins at the positive x-axis. Positive rotation moves toward the positive y-axis. This convention is common in science and engineering.
How the angle calculations work
The calculator first squares each Cartesian component. It adds those squared values. The square root gives the radius. Next, it compares z with the radius. The inverse cosine produces the inclination. Finally, atan2 uses y and x together. This produces the correct azimuth quadrant. Ordinary inverse tangent can lose that quadrant information. Therefore, atan2 is safer for vector conversion.
Choosing a compatible convention
Angle conventions can differ between textbooks. This page uses theta for inclination. Theta ranges from zero to 180 degrees. Phi represents azimuth. Phi can appear as a signed angle or a positive angle. Signed output ranges from minus 180 to 180 degrees. Positive output ranges from zero to 360 degrees. Both describe the same horizontal direction. Select the version that matches your coursework or software.
Zero vectors and precision
A zero vector has no direction. Its radius is zero. Inclination and azimuth cannot be uniquely defined. The calculator reports this clearly. For any nonzero vector, the angles are valid. Very small values may create rounding effects. Choose more decimal places when you need careful output. Use fewer places for readable reports.
Input checks that matter
Check signs before you calculate. A negative x component moves left. A negative y component changes the azimuth quadrant. A negative z component points below the horizontal plane. Those signs affect direction strongly. Do not replace them with absolute values. Enter the original vector components exactly.
Useful output beyond the angles
The result panel also shows normalized direction components. A unit vector has radius one. It preserves direction while removing length. This is helpful for comparing vectors of different sizes. You can export the calculated values as CSV. Save the printed result as PDF for records. The example table provides quick checks. Try a familiar vector first. Then use your required project values.
Applications in technical work
Coordinate conversion supports visualization. Spherical values make rotations easier to read. They also separate radial distance from orientation. This is valuable in simulation models. In graphics, direction guides cameras and lights. In robotics, it helps describe sensor targets. In electromagnetics, it helps compare field directions at many points.
Final comparison before sharing
Always confirm the target convention before sharing values. Some systems swap angle names. Others measure elevation from the xy-plane carefully.
Frequently asked questions
1. Which spherical convention does this calculator use?
It uses r for distance, θ for inclination from the positive z-axis, and φ for azimuth from the positive x-axis toward the positive y-axis.
2. What happens when all vector components are zero?
The radius becomes zero. Directional angles and normalized components are undefined because a zero vector has length but no direction.
3. Why does the calculator use atan2 for azimuth?
atan2 uses both x and y signs. It identifies the correct horizontal quadrant, unlike a simple inverse tangent calculation.
4. Can I calculate angles in radians?
Yes. Choose radians before converting. The calculator will return inclination, azimuth, and elevation in radians.
5. Are negative vector components allowed?
Yes. Negative components are important. They determine the vector direction and affect the azimuth and inclination values.
6. What is the difference between signed and positive azimuth?
Signed azimuth uses negative angles for clockwise directions. Positive azimuth reports the equivalent direction between zero and one full turn.
7. Is elevation the same as inclination?
No. Inclination starts at the positive z-axis. Elevation starts at the xy-plane. Together, they describe the same vertical direction differently.
8. How many decimal places should I select?
Use two to four places for ordinary work. Choose more places when calculations feed into engineering, scientific, or programming tasks.
9. Can x, y, and z use metres, feet, or other units?
Yes. Use one consistent length unit for all components. The radius will use that same unit. Angles stay unitless.
10. What are normalized direction components?
They are x, y, and z divided by radius. They form a unit vector that keeps direction while removing vector length.
11. How do I save my conversion results?
After calculating, use Download CSV for a spreadsheet file. Use Save Result as PDF to open your browser print dialog and save a PDF.