Direction of a Vector Calculator - Angles & Cosines

A vector's direction describes where it points in space, independently of its magnitude. While magnitude tells you how long or strong a vector is, direction tells you its orientation relative to the coordinate axes. The direction of a vector is most precisely expressed through direction angles — the angles that the vector makes with each positive coordinate axis — and through direction cosines, which are the cosines of those angles. For a 2D vector v = (x, y), the direction is typically given as a single angle α measured counterclockwise from the positive x-axis. The formula is α = arctan(y/x), but using the two-argument arctangent (atan2) ensures the correct quadrant is identified regardless of the signs of x and y. The direction cosines in 2D are cos α = x/|v| and cos β = y/|v|, where |v| is the magnitude √(x²+y²). For a 3D vector v = (x, y, z), there are three direction angles: α (angle with the x-axis), β (angle with the y-axis), and γ (angle with the z-axis). Each is computed as the arccosine of the corresponding direction cosine: cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|, where |v| = √(x²+y²+z²). A fundamental identity of direction cosines is that cos²α + cos²β + cos²γ = 1, which reflects the fact that the unit vector has length 1. The unit vector û in the direction of v is simply v divided by its magnitude: û = v/|v| = (x/|v|, y/|v|, z/|v|). It has a magnitude of exactly 1 and points in the same direction as v. Unit vectors are essential in physics and engineering for specifying directions without encoding any magnitude information — for example, the direction of a force, the orientation of a surface normal, or the pointing direction of a sensor. Direction calculations are fundamental in linear algebra, computer graphics, robotics, and physics. In 3D graphics, direction cosines and unit vectors define surface normals, lighting directions, and camera orientations. In robotics, they encode joint orientations and tool directions. In physics, forces, velocities, and field vectors all have directions that can be analyzed through direction angles. The calculator handles both 2D and 3D cases with full precision, computing all direction angles, direction cosines, the unit vector, and the magnitude in a single step.