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Vector Components Calculator

Enter a vector's magnitude and angle to find its x and y components.

Result

43.3013
25.0000

How it is calculated

Enter the vector's magnitude and its direction angle measured from the positive x-axis, counter-clockwise, with the unit option set to degrees (default) or radians. The tool resolves it into two perpendicular parts:

x = magnitude × cos(θ) and y = magnitude × sin(θ)

If the unit is degrees the angle is first converted with θ(rad) = degrees × π / 180. Results are rounded to four decimals and carry the same unit as the magnitude. Angles beyond 90° simply change the signs: x turns negative in the second and third quadrants, y in the third and fourth. Negative angles (clockwise) and angles above 360° are accepted.

Worked example

A force of 50 N acts at 30° above the horizontal. Convert: 30° = 0.5236 rad. Then x = 50 × cos 30° = 50 × 0.8660 = 43.3013 N and y = 50 × sin 30° = 50 × 0.5 = 25.0000 N. At 150° instead, the tool gives x = −43.3013 and y = 25.0000: same height, pointing to the left.

Try this example →

Common mistakes

  • Leaving the unit on degrees when the angle is in radians (or the reverse). 30 read as radians gives x = 7.7126 and y = −49.4016 instead of 43.3013 and 25.
  • Measuring the angle from the y-axis or from the nearest axis, not from the positive x-axis. A bearing of 30° from north is 60° in this convention.
  • Swapping the formulas, so that cos is used for the vertical part. That is correct only when the angle is measured from the vertical.
  • Ignoring the signs of the result for angles in the second, third and fourth quadrants.
  • Putting a negative number in the magnitude to indicate direction, which flips both components; keep the magnitude positive and use the angle for direction.

About this calculator

Any two-dimensional vector can be broken down into its horizontal (x) and vertical (y) components using basic trigonometry: given a magnitude |V| and a direction angle θ measured from the positive x-axis, the components are Vx = |V| × cos(θ) and Vy = |V| × sin(θ). This decomposition is the foundation of vector algebra, since components in the same direction can be added or subtracted directly, unlike vectors at arbitrary angles.

Resolving vectors into components is a core skill in physics coursework covering forces, velocity, and projectile motion, as well as engineering fields like structural analysis and mechanics, where a single force or displacement at an angle needs to be split along perpendicular axes before it can be combined with other forces or used in equilibrium equations.

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Comments (1)

  • Editorial team

    Components shine when several vectors act together. Resolve each one, add all the x values and all the y values separately, then find the resultant with Pythagoras. A 50 N force at 30 degrees and a 20 N force at 90 degrees give x = 43.3013 and y = 45, so the resultant is about 62.45 N at roughly 46.1 degrees. What problem are you resolving: forces, velocities or displacements?

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