Vector Direction Calculator - Direction in Both Units

A vector direction calculator returns the direction of a 2D vector from its x and y components, in degrees and radians, with the vector magnitude shown alongside. Type the x and y components, pick the angle range, and the result panel reads the direction angle theta that the vector makes with the positive x-axis, the matching angle in radians, and the matching magnitude. The math is atan2(y, x), and the tool handles every quadrant and both axes without a sign error.

Read the heading of a 2D velocity or force vector: An engineer with a velocity of 80 m east and 30 m north reads the heading as 20.56 degrees and the speed as 85.44 m in a single entry, without re-deriving the atan2 formula by hand.

Confirm the direction of a phasor in the I-Q plane: An electrical engineer with the in-phase and quadrature components of a phasor reads the direction angle in degrees or radians, the input every downstream mixing or demodulation step expects.

The direction of a vector is also called its heading, argument, or azimuth, but the math is the same: atan2(y, x), in the principal range (-180, 180] by default or the full range [0, 360) when positive-only angles are needed.

Readers who only need the size of the same (x, y) input can read the matching length on the Vector Magnitude Calculator without the angle math.

The tool reads the x and y components, computes the magnitude with the Pythagorean theorem, and recovers the direction angle with atan2(y, x). It multiplies the radian result by 180 over pi to get the degree value, and labels the quadrant from the signs of x and y. If the angle range is set to full, the tool wraps any negative result into [0, 360) degrees.

• x, y (vector components): The horizontal and vertical components of the 2D vector. Any real numbers are accepted, including 0 and negatives; the quadrant is determined by the sign combination.
• angleRange: Choice between the principal range (-180, 180] degrees (the convention used by Math.atan2 and most physics) and the full range [0, 360) (compasses, azimuths, robotics).
• theta (direction angle): The angle the vector makes with the positive x-axis, measured counterclockwise. Returned in both degrees and radians.
• |v| (magnitude): Length of the vector, computed as the square root of the sum of squared components. Always non-negative, equals 0 only for the zero vector.
• quadrant (label): Roman-numeral quadrant I, II, III, or IV from the signs of x and y, the axis label, or 'Origin' when both inputs are 0.

Math.atan2 is the standard two-argument inverse tangent built into JavaScript, Python, spreadsheets, and most scientific libraries. Its return value lives in (-pi, pi] radians, and the four sign combinations of (x, y) place the angle in Quadrant I, II, III, or IV, which is exactly the convention the calculator surfaces in the Quadrant row.

According to Weisstein, Eric W. "Inverse Tangent." MathWorld, atan2(y, x) returns the angle whose tangent is y/x and lies in (-pi, pi], using the sign of x to break ties and place the result in the correct quadrant. The same direction-angle formula appears in the OpenStax treatment of 2D vectors, where OpenStax Calculus Volume 3, Section 2.1 writes the direction angle of v = <x, y> as theta = arctan(y / x) and reminds the reader to apply the two-argument form so the signs of x and y place the angle in the right quadrant.

When you only need the angle from a y/x ratio rather than a full (x, y) point, the Arctan Calculator applies the same atan2-based quadrant logic on its arctan2 tab.

Four short ideas cover every direction-angle calculation you will run into, from a homework problem to a robotics heading.

When the problem also wants the radius r alongside the direction theta, the Rectangular to Polar Calculator returns both pieces from the same (x, y) input.

Five short steps take any 2D vector (x, y) to its direction angle in both units, including the trickier cases on the axes and at the origin.

1. 1 Enter the x and y components: Type the horizontal and vertical components of the vector. Positive, negative, zero, and decimals all work; any real pair is accepted.
2. 2 Choose the angle range: Pick Principal (-180 to 180 deg) for the programming-language convention, or Full (0 to 360 deg) for the compass or azimuth convention. The result panel updates to match the chosen range.
3. 3 Read the magnitude and direction: The first result row shows the vector magnitude |v|. The second and third rows show the direction angle in degrees and radians, in the range you selected.
4. 4 Check the quadrant label: The bottom result says I, II, III, IV, an axis label, or 'Origin' at (0, 0). It is the fastest way to confirm the answer is on the side of the plane you expected.
5. 5 Cross-check the direction: Square both x and y, add the squares, and confirm the result matches |v|^2. For any non-zero vector the magnitude in the panel is the length of the line from the origin to the tip.

A drone flies 80 meters east and 30 meters north from a field base. Type x = 80 and y = 30, keep the angle range on Principal, and the panel reads magnitude = 85.44, direction = 20.56 degrees / 0.3587 rad, quadrant = I. The drone's heading counterclockwise from due east is 20.56 degrees, which matches the asin(30/85.44) check by hand.

Readers who move from a 2D direction to a full 3D vector can take the (x, y) pair into the Vector Calculator and add a z component for the magnitude, unit vector, and three direction angles at once.

A short list of what the tool does well helps you put the result in the right place in your day.

• Both angle units in one step: Degrees and radians appear in the result panel at the same time, so there is no need to remember the 180/pi factor mid-calculation.
• • Atan2 used automatically: The two-argument arctangent picks the correct angle for every (x, y) input, including the four quadrants and the two axes where the single-argument arctan would silently return 0.
• • Principal and full range supported: A toggle between the (-180, 180] and [0, 360) conventions covers both the programming-language default and the compass or azimuth convention without a second manual step.
• • Origin handled explicitly: When both x and y are zero, the calculator returns magnitude 0 and an explicit 'Undefined' direction rather than a silent NaN.

Readers who only need the acute reference angle rather than the full direction can take the same (x, y) input into the Reference Angle Calculator and read the acute angle to the nearest x-axis.

Three things change the answer you should expect, plus two practical caveats about how vector direction is usually defined.

• • The direction of a 2D vector is unique up to adding or subtracting full turns of 360 degrees. The principal range gives the unique value in (-180, 180] and the full range gives the unique value in [0, 360); both describe the same physical direction.
• • The angle is rounded to 4 decimal places in degrees and 6 in radians. That is enough for engineering and academic work, but high-precision metrology may want to extend the displayed precision.

According to Wikipedia, "Atan2", atan2(y, x) is the angle in the Euclidean plane, measured counterclockwise, between the positive x-axis and the point (x, y), with a range of (-pi, pi] radians

Readers who need the direction of a sum of two 2D vectors can add the components with the Vector Addition Calculator and then feed the result into the same direction formula on this page.