Vector A (v)
Vector B (u)
Visualize Vector Addition
Our interactive calculator simplifies vector mathematics. Input your components and see the results calculated and graphed in real-time.
Input Vector A
Enter the X₁ and Y₁ components for the first vector (v). The graph will instantly update to show its position.
Input Vector B
Enter the X₂ and Y₂ components for the second vector (u). Watch as it appears on the graph alongside the first vector.
Analyze Results
The resultant vector (v+u) is automatically calculated and drawn, with its components, magnitude, and angle displayed below.
Advanced Calculation & Visualization
Go beyond simple numbers. Our tool provides a dynamic graphical representation, making complex concepts intuitive and easy to understand.
- Live Cartesian Graph: All vectors are plotted instantly on a 2D plane, providing immediate visual feedback for your inputs.
- Comprehensive Analysis: Automatically computes the resultant vector's components, its magnitude (length), and its direction (angle) in degrees.
- Parallelogram Law Visualization: The graph illustrates the "tip-to-tail" or parallelogram method of vector addition, making the geometric relationship clear.
- Adaptive Scaling: The graph's coordinate system dynamically scales to optimally fit vectors of any magnitude, from small to very large values.
The Principles of Vector Addition
Vectors are fundamental in physics, engineering, and computer graphics. Understand the core concepts behind the calculations our tool performs.
How are vectors added algebraically?
To add two vectors, you simply add their corresponding components. If you have Vector A = (x₁, y₁) and Vector B = (x₂, y₂), their sum, the resultant vector R, is calculated as:
R = (x₁ + x₂, y₁ + y₂)
Our calculator performs this component-wise addition instantly.
What is the Parallelogram Law?
The Parallelogram Law is a geometric way to add vectors. If you draw two vectors starting from the same origin, they form two adjacent sides of a parallelogram. The resultant vector is the diagonal of this parallelogram that also starts from the same origin. Our visualization graph demonstrates this principle by drawing the helper lines to complete the shape.
How are Magnitude and Angle calculated?
For a resultant vector R = (Rx, Ry):
- Magnitude: This is the length of the vector. It's calculated using the Pythagorean theorem:
Magnitude = √(Rx² + Ry²). - Angle (Direction): This is the angle the vector makes with the positive x-axis. It's found using the inverse tangent function:
Angle = atan2(Ry, Rx). Theatan2function is used to get the correct angle in all four quadrants.
