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Vector Components Calculator
Convert a vector between polar form and Cartesian components. This tool shows how magnitude, direction, x-component, and y-component are connected through trigonometry, coordinate geometry, and physics modeling.
Core relationships
x = r cos θ
y = r sin θ
r = √(x² + y²)
θ = atan2(y, x)
Calculator
Use component mode when you know x and y. Use polar mode when you know magnitude r and direction θ.
4.90
3.40
Magnitude r
5.97
Angle θ
34.7°
x-component
4.90
y-component
3.40
Quadrant
I
Slope y/x
0.69
Current calculation
x = r cos θ, y = r sin θ
Vector r
x and y projections
Angle θ
What this calculator teaches
A vector can be represented in two equivalent forms. Cartesian form describes the vector using horizontal and vertical components. Polar form describes the same vector using a magnitude and a direction.
Formulas and when to use them
Use the polar-to-Cartesian formulas when the vector is given by magnitude and angle. Use the Cartesian-to-polar formulas when the vector is given by components.
x = r cos θ
y = r sin θ
r = √(x² + y²)
θ = atan2(y, x)
Angle conventions and quadrants
The sign of each component depends on the quadrant. This is one of the most important ideas in vector decomposition.
Worked examples
Special cases and edge cases
Real-world uses of this calculator
The same conversion appears whenever a directional quantity must be separated into perpendicular effects.
Common mistakes and how to avoid them
Interactive practice generator
Generate a random vector problem, solve it by hand, then reveal the answer.
A vector has r = 8 and θ = 45°. Find x and y.
Hint: use x = r cos θ and y = r sin θ.
x ≈ 5.66, y ≈ 5.66
Additional practice set
Glossary
FAQ
Latest Resources
Recently published tools and explanations.
Vector Components from Polar Coordinates
Interactive Physics Tool English Tiếng Việt Español Français 日本語 한국어 中文 Calculator Concept Formulas Examples Practice Mistakes FAQ Vector Components Calculator Convert a vector between polar…