How do you calculate vector magnitude?

For a 2D vector, the magnitude is sqrt(x squared plus y squared). For a 3D vector, the magnitude is sqrt(x squared plus y squared plus z squared).

How do you find the angle of a vector?

In 2D, the direction angle from the positive x-axis is found with atan2(y, x). In 3D, direction angles can be found from acos(x over magnitude), acos(y over magnitude), and acos(z over magnitude).

AmesWeb

Vector Calculator

Q: Can this calculator find vector components from magnitude and angle?

Yes. In 2D it converts magnitude and angle into x and y components. In 3D it converts magnitude and direction angles into x, y, and z components.

Calculate vector magnitude, vector angle, vector direction, and vector components for 2D and 3D vectors. This free vector calculator converts between component form and magnitude with angle.

Vector Magnitude, Angle, Direction and Components Calculator

Use this page as a vector magnitude calculator, vector angle calculator, vector direction calculator, vector component calculator, component form calculator, or 3D vector calculator.

Calculator Inputs

Input mode

Vector dimension

Vector Components

x component

y component

z component

Magnitude and Direction Angle(s)

|A|

θx / θ (deg)

θy (deg)

θz (deg)

Use “.” as the decimal separator.

Results

|A| = √(Ax² + Ay²)

x component

—

y component

—

z component

—

Magnitude |A|

—

Direction angle θx / θ

—

Direction angle θy

—

Direction angle θz

—

What this means

Enter values and click Calculate to convert between vector components and magnitude with direction angles.

2D Vector Diagram

For 2D vectors, the calculator also draws the vector from the origin and shows its x and y components.

Vector Formulas

2D magnitude: |A| = √(Ax² + Ay²)

3D magnitude: |A| = √(Ax² + Ay² + Az²)

2D direction angle: θ = atan2(Ay, Ax)

3D direction angles: θx = cos⁻¹(Ax/|A|), θy = cos⁻¹(Ay/|A|), θz = cos⁻¹(Az/|A|)

How to Use the Vector Calculator

Select Vector Components if you know x, y, and optionally z.
Select Magnitude and Angle if you know the vector size and direction.
Choose 2D or 3D mode.
Enter the known values.
Click Calculate to get magnitude, angle, direction, and component form.

Examples

Case	Input	Use case
2D vector	A = <5, -6>	Find magnitude and angle of a vector
3D vector	A = <3, 4, 12>	Use as a 3D vector calculator
2D magnitude-angle	\|A\| = 10, θ = 30°	Find x and y components of a vector
3D magnitude-angle	\|A\| = 15, θx = 60°, θy = 45°, θz = 60°	Convert direction angles to component form

About This Vector Calculator

This tool calculates vector magnitude, direction angle, and vector components for both 2D and 3D vectors. It can also convert between component form and magnitude-angle form, making it useful for common vector calculations in physics, mathematics, and engineering.

FAQ

How do you calculate the magnitude of a vector?

For a 2D vector, use √(x² + y²). For a 3D vector, use √(x² + y² + z²).

How do you find the direction angle of a vector?

In 2D, use atan2(y, x). In 3D, direction angles come from the direction cosines.

Can this calculator find vector components from magnitude and angle?

Yes. In 2D, it returns x and y components. In 3D, it returns x, y, and z components.

Is this also a component form calculator?

Yes. It converts between component form and magnitude-angle form for 2D and 3D vectors.

Related Calculators

Physics Calculators Dot Product Calculator Vector Addition Calculator Vector Cross Product Calculator 2D Vector Addition Calculator

References

Giancoli, Douglas C. Physics: Principles with Applications. Pearson.
Halliday, Resnick, Walker. Fundamentals of Physics.
OpenStax. University Physics. Vector fundamentals and component methods.