Overview
A vector is a quantity with magnitude and direction, represented in the plane by horizontal and vertical components. Its magnitude is the distance from tail to head, and its unit vector points in the same direction with length 1.
Formula
|v| = √(x² + y²), and u = (x/|v|, y/|v|).
How it works
Enter the components
x horizontal and y vertical.
Compute the magnitude
Apply the Pythagorean theorem.
Compute the direction
Use atan2(y, x), taking the quadrant into account.
Divide by the magnitude
To obtain the unit vector.
Worked examples
Example 1: (3, 4)
Magnitude 5, direction ≈ 53.1301°, unit vector (0.6, 0.8).
Example 2: (−3, 3)
Magnitude ≈ 4.2426, direction 135°.
Frequently asked questions
What is the difference between a vector and a scalar?
A vector has direction and magnitude, while a scalar has magnitude only.
Can a vector have negative magnitude?
No, magnitude is always nonnegative.
How do I find the direction in the third quadrant?
Use atan2, which returns the correct angle between −180° and 180°.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.