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.
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°.
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.
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.