ConverterHub

2D Vector Calculator

Find the magnitude and angle of a 2D vector.

How it is calculated

The magnitude is |v| = √(x² + y²), computed with the hypotenuse function and shown to 4 decimals. The direction is θ = atan2(y, x) converted to degrees and shown to 2 decimals. Unlike a plain arctan(y/x), atan2 uses the signs of both components, so the angle lands in the correct quadrant and runs from −180° to 180°, measured counter-clockwise from the positive x axis. A vector with x = 0 and y = 0 has magnitude 0 and the angle is reported as 0°.

Worked example

Take the vector (3, 4). Magnitude: 3² + 4² = 9 + 16 = 25, and √25 = 5.0000. Angle: atan2(4, 3) is the angle whose tangent is 4/3, which is 53.13° above the positive x axis.

Try this example →

Common mistakes

  • Computing arctan(y / x) by hand for negative x. A vector at (−3, 4) points at about 126.87°, not −53.13°.
  • Expecting an angle between 0° and 360°. Results below the x axis are negative, so −45° means 315°.
  • Swapping x and y, which mirrors the angle around 45°.
  • Reading the angle as a compass bearing. Bearings start at north and turn clockwise, while this angle starts at east and turns counter-clockwise.

About this calculator

This calculator finds a 2D vector's magnitude, |v| = √(x² + y²) (from the Pythagorean theorem), and its direction angle, θ = arctan(y/x) (adjusted for the correct quadrant), given its x and y components. Together, magnitude and angle describe the same vector in polar form instead of Cartesian form — the two representations are mathematically interchangeable.

Physics and engineering students use this to convert between component and polar notation for forces, velocities and displacements, and it's a basic building block in computer graphics, robotics and game development for representing direction and speed. Enter the x and y components to get the magnitude and angle.

Was this helpful?

You can pick up to 3 reactions.

Comments (0)

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators