How to use
Add two to ten 2D vectors using magnitudes and angles or x/y components. See the resultant, live diagram, direction formats and calculation steps.
Choose Magnitude + Angle or X / Y Components. Every row uses that format. Enter two to ten vectors in the same compatible quantity unit and coordinate frame; labels run A through J. Add Vector adds a zero row. Remove is disabled when two rows remain. Valid values convert when switching format; correct invalid fields before switching. For a zero vector, a converted polar input angle of zero is only a placeholder and does not define its direction.
Choose Degrees or Radians, Standard Angle, Compass or Bearing, and Auto or 2, 3, 4 or 6 decimal places. Results update in page memory. Copy an individual magnitude, direction or component, or use Copy Result for the accepted vectors and resultant. Clipboard failure provides selectable text. Load Example offers perpendicular, opposing and three-vector fixtures in your current input format and angle unit. Reset restores A = 10 @ 30°, B = 7 @ 120°, polar input, degrees, standard direction and Auto precision, and closes Show steps. Refresh clears edits.
Vector addition basics
A vector has magnitude and direction. Two-dimensional vector addition combines horizontal and vertical effects in the same coordinate system. The sum R is the resultant. Source arrows can be placed at a shared origin for comparison; component sums still implement the same head-to-tail addition rule.
The diagram draws all source vectors and R from the origin, with positive x right and positive y up. Its uniform automatic scale preserves relative lengths and angles. Labels, dashed source lines and the thicker solid resultant distinguish vectors without relying only on color.
Magnitude and angle
For magnitude r ≥ 0 and standard angle θ, x = r cos θ and y = r sin θ. At 0°, the vector points right; at 90°, up; at 180°, left; and at 270°, down. Negative angles and multiple revolutions are accepted within the computational bound. For example, −30° is equivalent to 330°, and 450° to 90°.
A zero magnitude gives x = y = 0 regardless of its entered angle. Its direction is undefined, so the result and breakdown explicitly say “Direction undefined for zero magnitude.”
X/Y components
Component input supplies the signed horizontal x and vertical y values directly. Positive x points right and negative x left; positive y points up and negative y down. No trigonometric decomposition is needed in this mode. Each vector’s magnitude and standard direction are derived for the breakdown.
All quantities must represent compatible units, such as all forces in newtons or all displacements in metres. The calculator does not infer units, convert between incompatible quantities, or combine vectors from different coordinate frames.
Adding components and formulas
Rx = Σx and Ry = Σy. Add all signed x-components, then all signed y-components. For A = (5, −2) and B = (−1, 7), Rx = 5 + (−1) = 4 and Ry = −2 + 7 = 5. Component summation uses compensated arithmetic to retain small terms during large-value cancellation.
Resultant magnitude
R = √(Rx² + Ry²). This is the Pythagorean length of the resultant, not the sum of the individual magnitudes. The implementation uses Math.hypot for stable length calculations. Perpendicular vectors of magnitudes 3 and 4 produce magnitude 5; opposite vectors of equal magnitude cancel.
You can verify the magnitude from perpendicular components in a separate right-triangle model, using their nonnegative lengths in consistent units. This checks magnitude only; retain component signs for the vector direction and coordinate convention.
Resultant direction
θ = atan2(Ry, Rx). The two-argument arctangent retains both component signs, selecting the correct quadrant and handling vertical vectors without division by zero. Normalize its result into 0° ≤ θ < 360° or 0 ≤ θ < 2π radians. Zero resultants have no direction.
Angles very close to a full revolution can round to zero in the displayed cyclic format; the full internal direction is retained. Display precision controls rounding, not the underlying computation or the accuracy of measurements.
Direction conventions
Standard angle
Standard angle increases counterclockwise from the positive x-axis: East 0°, North 90°, West 180°, South 270°. It uses the selected degree/radian unit.
Compass azimuth
Azimuth = (90° − standard angle) mod 360°. It increases clockwise from North: N 0°, E 90°, S 180°, W 270°. The number is the azimuth; an N, NE, E, SE, S, SW, W or NW label is only the nearest eight-point compass direction. Halfway ties go clockwise. Compass values always use degrees.
Quadrant bearing
Quadrant bearings start at North or South and turn toward East or West, such as N 36.87° E or S 36.87° W. Exact cardinal directions show only N, E, S or W. For R = (3, 4), standard angle is about 53.1301°, compass azimuth 36.8699° NE, and bearing N 36.8699° E. Bearing angles always use degrees.
Degrees vs radians
A full turn is 360° or 2π radians. Convert radians to degrees with θdeg = θrad × 180/π, and degrees to radians with θrad = θdeg × π/180. Changing the angle-unit selector converts entered angles, preserving their direction rather than turning 180 degrees into 180 radians. Angles are normalized before trigonometric evaluation.
Auto uses up to six significant digits. Fixed options use the chosen decimal places; very small nonzero values fall back to scientific notation to avoid a false zero, and very large values use scientific notation to keep output readable. These settings describe formatting, not significant measurement precision.
Worked examples
Perpendicular 3–4–5 vectors
A = 3 @ 0° = (3, 0); B = 4 @ 90° = (0, 4). R = (3, 4), magnitude √(3² + 4²) = 5 and direction atan2(4, 3) ≈ 53.1301023542°.
The default fixture
A = 10 @ 30° gives Ax = 8.660254037844387 and Ay ≈ 5. B = 7 @ 120° gives Bx ≈ −3.5 and By = 6.062177826491071. Rx ≈ 5.160254037844389 and Ry = 11.06217782649107. The magnitude is √149 = 12.206555615733702; direction is 30° + atan2(7, 10) = 64.99202019855866°. Auto displays 12.2066 and 64.992°. These are derived values, not claims of input measurement accuracy.
Direct components
A = (5, −2), B = (−1, 7). R = (4, 5), magnitude √41 ≈ 6.403124237432848, direction atan2(5, 4) ≈ 51.34019174590991°.
Equal and opposite cancellation
A = 10 @ 0°, B = 10 @ 180°. R = (0, 0), magnitude zero, direction undefined. Equal and opposite vectors cancel.
Three-vector example
A = 3 @ 0°, B = 4 @ 90°, C = 3 @ 180°. The horizontal components cancel; R = (0, 4), magnitude 4 and standard direction 90°.
Assumptions and limitations
This is a 2D addition calculator for vectors in one coordinate frame and compatible quantity units. It does not model 3D vectors, dot products, cross products, matrices, vector fields, changing coordinate frames, uncertainty propagation or physical forces beyond the supplied vector data.
Each numeric field accepts at most 64 characters and finite decimal/scientific syntax. Each nonzero magnitude or component must have absolute value at least 1e-100. Each vector’s magnitude is bounded by 1e12, including the combined magnitude of component inputs; negative polar magnitudes are rejected. Angles are bounded to one million revolutions in either direction. There are two to ten rows. Resultant magnitude cannot exceed 1e13. These limits keep work and arithmetic bounded.
Numerical zero tolerance = 8 × Number.EPSILON × Σ(input magnitudes), where Number.EPSILON ≈ 2.220446049250313e-16. A resultant within this scale-relative tolerance is classified as zero, with raw magnitude and tolerance disclosed. Individual tiny vectors are not subject to an absolute zero cutoff. IEEE-754 rounding remains possible; the classification is numerical, not a statement about measurement uncertainty.
FAQ
How do I add two vectors?
Express both vectors in the same x/y coordinate system and compatible unit. Add their x-components and their y-components separately. The resulting pair is the resultant vector.
How do I add vectors using magnitude and angle?
Resolve each magnitude r and standard angle θ into x = r cos θ and y = r sin θ. Add the components, then calculate the resultant magnitude and direction.
How do I convert a vector into x and y components?
Use x = r cos θ and y = r sin θ, with θ measured counterclockwise from the positive x-axis. The signs of the components identify the quadrant.
How do I find the magnitude of the resultant vector?
Use R = √(Rx² + Ry²). The implementation uses Math.hypot(Rx, Ry), a stable way to calculate the same length.
How do I find the direction of the resultant?
Use θ = atan2(Ry, Rx) and normalize the angle to one revolution. Direction is undefined if the resultant is zero within the documented numerical tolerance.
Why does the calculator use atan2 instead of ordinary arctangent?
atan2 uses both signed components, preserving the correct quadrant and handling a zero x-component. atan(Ry/Rx) loses quadrant information and divides by zero on vertical vectors.
Can I add more than two vectors?
Yes. Add up to ten vectors, labeled A through J. At least two input rows remain; use a zero vector if only one nonzero vector is needed.
Can I use radians instead of degrees?
Yes. Changing Angle unit converts existing valid polar angles instead of reinterpreting their numbers. In component mode, it changes only the standard-angle display. Compass azimuth and quadrant bearings always use degrees.
What happens if the resultant vector is zero?
Its magnitude and components display zero, and its direction is undefined. Equal and opposite vectors cancel. A scale-relative floating-point tolerance recognizes negligible cancellation residuals without applying an absolute cutoff to small vectors.
What is the difference between standard angle, compass direction, and bearing?
Standard angle increases counterclockwise from +x. Compass azimuth increases clockwise from North. A quadrant bearing starts from North or South and gives an acute angle toward East or West; exact cardinal directions display N, E, S or W.
Sources
Reviewed October 1, 2026. OpenStax University Physics Volume 1 explains coordinate components, vector magnitudes, planar directions and component-based vector addition. Compass/bearing formatting and computational bounds are documented product conventions. Sources do not endorse BuzCalculator.