Enter your vector's X component and Y component to find the direction angle (θ) it makes with the positive x-axis, plus its magnitude and unit vector. The Vector Direction Calculator handles all four quadrants and returns angles in both degrees and radians. Also try the Dot Product Calculator.
Direction Angle (θ)
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Direction Angle (Radians)
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Magnitude ‖v‖
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Unit Vector X (û₁)
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Unit Vector Y (û₂)
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The direction of a 2D vector v = (x, y) is the angle θ it makes with the positive x-axis. You can find it using θ = arctan(y / x), but you must account for the quadrant the vector lies in. For vectors in the second or third quadrant, add 180°; for the fourth quadrant, add 360° to keep the angle in the range 0° to 360°. See also our use the Polar Coordinates Calculator.
To find the direction angle, use the formula θ = atan2(y, x), which automatically handles all four quadrants. The result is in radians; multiply by 180/π to convert to degrees. This calculator does all of that for you automatically and returns the angle in both degrees and radians.
A unit vector û in the direction of v = (x, y) is found by dividing each component by the magnitude: û = (x / ‖v‖, y / ‖v‖), where ‖v‖ = √(x² + y²). The resulting vector has the same direction as v but a magnitude of exactly 1.
The magnitude ‖v‖ of a vector v = (x, y) is its length, calculated using the Pythagorean theorem: ‖v‖ = √(x² + y²). It is always a non-negative value representing how long the vector is, independent of its direction. You might also find our calculate Scalar Projection (comp), Projection Vector — x & Projection Vector — y — Vector Projection useful.
Because the direction angle depends only on the ratio y/x. If two vectors like u = (-2, 3) and v = (-4, 6) share the same ratio, arctan(y/x) returns the same angle. Scaling a vector changes its magnitude but not its direction.
If x = 0, the vector points purely along the y-axis. The direction angle is 90° (π/2 radians) if y > 0, and 270° (3π/2 radians) if y < 0. The arctan formula is undefined at x = 0, so the atan2 function is used instead to handle this case correctly.
The dot product of two vectors pointing in the same direction is positive. If the angle θ between them is 0°, then the dot product equals the product of their magnitudes (a positive number). Vectors pointing in opposite directions have a negative dot product.
To find the resultant of two vectors, first add their corresponding components: (x₁ + x₂, y₁ + y₂). Then calculate the magnitude of the summed vector using √((x₁+x₂)² + (y₁+y₂)²) and its direction using atan2(y₁+y₂, x₁+x₂). Each vector can be entered separately into this calculator to check individual directions.