Vector Magnitude Calculator with Angle

Solve vectors with components, points, or polar input. See magnitude, angle, unit vector, and quadrant. Export results, inspect plots, and verify answers with examples.

Vector Magnitude and Angle Calculator

Choose a mode, enter known values, and calculate the vector magnitude, direction, unit vector, and related measures.

This mode finds magnitude and direction from the entered rectangular components.
This mode converts polar input into vector components using the selected axis and rotation.
This mode builds the vector B − A, then computes its size and direction.

Example Data Table

Case Input Type Given Values Vector Magnitude Angle from +X
Example 1 Components x = 3, y = 4 ⟨3, 4⟩ 5.0000 53.1301°
Example 2 Components x = -5, y = 12 ⟨-5, 12⟩ 13.0000 112.6199°
Example 3 Polar Magnitude = 10, Angle = 30° ⟨8.6603, 5.0000⟩ 10.0000 30.0000°
Example 4 Points A(1, 2), B(6, 5) ⟨5, 3⟩ 5.8310 30.9638°

Formula Used

Magnitude from components: |v| = √(x² + y²)

Angle from components: θ = atan2(y, x)

Components from magnitude and angle: x = r cos θ, y = r sin θ

Vector from points: v = (x₂ − x₁, y₂ − y₁)

Unit vector: û = v / |v|

The atan2 function is important because it places the angle in the correct quadrant automatically. That avoids sign mistakes when X or Y is negative.

How to Use This Calculator

  1. Select a calculation mode based on the values you already know.
  2. Choose the reference axis and the rotation direction for the angle.
  3. Enter the components, polar values, or point coordinates.
  4. Set the number of decimal places for the displayed result.
  5. Press Calculate Vector to show the result above the form.
  6. Review the magnitude, angle, unit vector, slope, quadrant, and graph.
  7. Use the CSV button for spreadsheet work or the PDF button for printable output.

FAQs

1) How do you find vector magnitude from components?

Square the X and Y components, add them, and take the square root. For ⟨3, 4⟩, the magnitude is √(3² + 4²) = 5.

2) How is the angle of a vector usually measured?

The standard convention measures the angle from the positive X-axis, moving counterclockwise. This calculator also lets you switch the reference axis and rotation direction.

3) Why is atan2 better than arctan(y/x)?

atan2 uses both component signs, so it identifies the correct quadrant. A plain arctan(y/x) can return an angle that points the wrong way.

4) What happens for the zero vector?

Its magnitude is zero, but its angle is undefined because it has no direction. The unit vector is also undefined in that special case.

5) Can a vector have a negative component and still a positive magnitude?

Yes. Magnitude is always nonnegative because it comes from squared components. Negative X or Y values only affect direction and quadrant.

6) When should I use degrees or radians?

Degrees are common for classroom and engineering work. Radians are preferred in calculus, physics, and trigonometric identities. This calculator shows both outputs.

7) Can two points define a vector?

Yes. Subtract the starting point from the ending point. That gives the vector components, which then determine magnitude and direction.

8) What is a unit vector?

A unit vector has magnitude one and shows direction only. You get it by dividing each component by the vector magnitude.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.