Choose a vector or bearing task
Core formulas used
Components from bearing
North = r × cos(θ)
Bearing from components
θ = (θ + 360°) mod 360°
Magnitude
Dot product
Projection
Great-circle distance
How to use this calculator
Select a calculation mode that matches your available data. Enter components, bearings, coordinates, or multiple vector rows. Choose the correct angle and distance units before calculating.
Review the result cards, detailed steps, diagram, and table. Use CSV export for spreadsheets. Use Print or PDF to create a formatted report from your browser.
For navigation data, verify coordinate signs and the bearing convention. For magnetic corrections, use a current declination value. For survey work, inspect misclosure before applying adjustments.
Calculation history
Bearing conventions and practical notes
A navigation bearing starts at north and increases clockwise. A mathematical direction angle starts at east and increases counterclockwise. Converting between them requires a ninety-degree axis rotation and normalization.
Quadrant bearings use north or south first, followed by an acute angle and east or west. Whole-circle bearings avoid quadrant ambiguity. Back bearings point exactly opposite the original direction.
Vector components preserve both magnitude and direction. East and north components are especially useful for surveying, aviation, marine navigation, GIS, robotics, and engineering force analysis.
Geographic calculations treat Earth as a sphere in this standalone version. Great-circle routes represent shortest spherical paths. High-precision geodesy may require an ellipsoidal library and a selected datum.
Magnetic declination varies by position and date. Grid convergence depends on the map projection. Never treat an old correction value as universally valid.