Direction of the Vector Calculator

In analytical geometry, the direction of a two-dimensional vector v = (x, y) is typically defined by the angle θ it forms with the positive x-axis. This is calculated using the inverse tangent function: θ = atan2(y, x). To find the unit vector pointing in this exact same direction, each component of the vector is divided by its overall magnitude, where the magnitude is computed using the Pythagorean theorem: |v| = √(x² + y²). For three-dimensional vectors involving x, y, and z coordinates, magnitude expands to |v| = √(x² + y² + z²), and directional properties are derived by computing directional cosines or normalized unit vector components.

Let us find the magnitude, angle, and unit vector for a 2D vector where x = 3 and y = 4. First, calculate the magnitude using the formula |v| = √(3² + 4²), which equals √(9 + 16) = √(25) = 5. Next, determine the directional angle using θ = atan2(4, 3), resulting in approximately 53.13 degrees. Finally, to find the unit vector pointing in this same direction, divide each component by the magnitude: x̂ = 3 / 5 = 0.6 and ŷ = 4 / 5 = 0.8. Thus, the unit vector is (0.6, 0.8).

Always double-check your coordinate signs (+/-), as they dictate which quadrant the vector angle falls into. When working with trigonometric functions like atan2, ensure your calculator is set to the correct mode (degrees or radians) depending on your assignment requirements. If you need to scale a vector to a specific target magnitude without changing its path, first reduce it to a unit vector, then multiply every component by your desired scalar length.