About

Miscalculating vector components in structural analysis, navigation, or game physics leads to cascading errors in force resolution and trajectory prediction. This tool computes 2D vectors from either magnitude and angle (r, θ) or Cartesian components (x, y), resolves the resultant via component summation, and renders a publication-ready diagram. All trigonometric decomposition uses IEEE 754 double-precision floating point. The resultant is calculated as R = √R_x² + R_y², which is exact for any number of concurrent coplanar vectors. Export produces a standards-compliant SVG file suitable for embedding in technical documents or further editing in Inkscape or Illustrator.

Limitation: this tool handles 2D coplanar vectors only. It assumes all vectors originate from a common point (concurrent). Non-concurrent force systems require moment calculations not covered here. Angle input follows the mathematical convention: 0° points along the positive x-axis, increasing counterclockwise. Pro tip: when resolving forces on an inclined plane, rotate your coordinate system first so one axis aligns with the surface - it reduces the number of vectors you need.

Formulas

Each vector v is defined by its magnitude r and direction angle θ measured counterclockwise from the positive x-axis. The Cartesian components are decomposed as:

v_x = r cos θ
v_y = r sin θ

Conversely, given components (x, y):

r = √x² + y²
θ = atan2(y, x)

The resultant of n vectors is computed by summing all components and then converting back:

R_x = n∑i=1 v_{x_i}
R_y = n∑i=1 v_{y_i}
R = √R_x² + R_y²

Where r = vector magnitude, θ = direction angle in degrees (converted to radians internally via θ_rad = θ × π180), v_x and v_y = horizontal and vertical components, R = resultant magnitude, R_x and R_y = summed components of all vectors.

Reference Data

Quantity	Symbol	Formula / Definition	Unit
Magnitude	r	√x² + y²	any
Direction Angle	θ	atan2(y, x)	°
X-Component	x	r cos θ	any
Y-Component	y	r sin θ	any
Resultant Magnitude	R	√(∑x_i)² + (∑y_i)²	any
Resultant Angle	θ_R	atan2(∑y_i, ∑x_i)	°
Unit Vector	u	v\|v\|	dimensionless
Dot Product	a ⋅ b	a_xb_x + a_yb_y	scalar
Cross Product (2D)	a × b	a_xb_y − a_yb_x	scalar (z-component)
Angle Between Two Vectors	α	acos(a ⋅ b\|a\| \|b\|)	°
Projection of a onto b	proj	a ⋅ b\|b\|	any
Common angle: East	-	0°	-
Common angle: North	-	90°	-
Common angle: West	-	180°	-
Common angle: South	-	270°	-
Common angle: NE diagonal	-	45°	-
Common angle: NW diagonal	-	135°	-

Frequently Asked Questions

Angles follow standard mathematical convention: 0° points along the positive x-axis (right/east), and angles increase counterclockwise. This differs from navigation bearings where 0° is north and increases clockwise. If you need navigation bearings, convert by computing θ_math = 90 − θ_bearing.

The component summation method is numerically stable. Even when two large vectors nearly cancel (e.g., magnitudes of 1000 at 0° and 999 at 180°), the resultant is computed to full IEEE 754 double-precision (≈15 significant digits). Catastrophic cancellation is negligible for practical use.

Yes. The SVG output uses standard elements (<line>, <polygon>, <text>) with no proprietary attributes. It imports cleanly into Inkscape, Adobe Illustrator, Figma, and LaTeX (via the svg package). All coordinates are unitless and map to the canvas pixel space; scale as needed in your target application.

A zero-magnitude vector is mathematically valid - it is the zero vector 0 with undefined direction. The tool accepts it, assigns the specified angle (which becomes irrelevant), and draws a point at the origin. It contributes nothing to the resultant. This is useful as a placeholder when building a system incrementally.

The atan2 function returns the full 360° angle, accounting for the correct quadrant. A naive atan(y÷x) only resolves to −90° to 90° and fails in quadrants II and III. If your resultant has a negative R_x, the angle will correctly land between 90° and 270°.

There is no hard mathematical limit. The canvas rendering and SVG export handle up to around 50 vectors comfortably. Beyond that, labels may overlap and the diagram becomes visually cluttered. The resultant calculation itself is a simple linear summation and would handle thousands without performance issues.