Vector Addition Calculator – Guide & Formulas
Add two or more vectors component-wise. Get the resultant vector magnitude and direction for 2D or 3D vector addition.
Add two or more vectors component-wise to find the resultant vector. Get the magnitude and direction of the resultant for both 2D and 3D vector addition.
Key Takeaway
Use the free Vector Addition Calculator to add two or more vectors component-wise. get the resultant vector magnitude and direction for 2d or 3d vector addition. Get instant results with step-by-step explanations.
How to Use the Vector Addition Calculator
- Enter the components of the first vector (Ax, Ay, Az).
- Enter the components of the second vector (Bx, By, Bz).
- Optionally add more vectors to sum.
- Click "Calculate" to see the resultant vector magnitude and direction.
The Formula
Variable Definitions
- R⃗: Resultant vector (sum of all vectors)
- A⃗, B⃗: Input vectors being added
- Ax, Ay, Az: Components of vector A
- Bx, By, Bz: Components of vector B
- |R⃗|: Magnitude of the resultant vector
Example: Add Vectors A = (3, 4) and B = (1, 2)
Find the resultant of adding vectors A = (3, 4) and B = (1, 2).
- Step 1: Add x-components: Rx = 3 + 1 = 4.
- Step 2: Add y-components: Ry = 4 + 2 = 6.
- Step 3: Resultant vector R = (4, 6).
- Step 4: Calculate magnitude: |R| = √(4² + 6²) = √(16 + 36) = √52 ≈ 7.211.
- Step 5: Calculate direction: θ = arctan(6/4) = arctan(1.5) ≈ 56.31°.
Frequently Asked Questions
What is vector addition?
Vector addition combines two or more vectors by adding their corresponding components. The result is a new vector called the resultant.
Is vector addition commutative?
Yes, vector addition is commutative: A⃗ + B⃗ = B⃗ + A⃗. The order of addition does not affect the result.
What is the parallelogram law?
The parallelogram law states that if two vectors are placed tail-to-tail, their sum is the diagonal of the parallelogram formed by the two vectors.
What is the head-to-tail method?
In the head-to-tail method, place the tail of the second vector at the head of the first. The resultant goes from the tail of the first to the head of the last.
How do I subtract vectors?
Vector subtraction is adding the negative: A⃗ - B⃗ = A⃗ + (-B⃗). Reverse the direction of B and then add.
Can I add more than two vectors?
Yes, you can add any number of vectors. Simply add all x-components together, all y-components together, and all z-components together.
What is the magnitude of the resultant?
The magnitude is calculated using the Pythagorean theorem: |R⃗| = √(Rx² + Ry² + Rz²). It represents the length of the resultant vector.
How do I find the direction of the resultant?
For 2D vectors, the direction angle is θ = arctan(Ry/Rx). Adjust the angle based on the quadrant of the resultant.
What is the zero vector?
The zero vector (0, 0, 0) has zero magnitude and no specific direction. It is the result of adding a vector and its negative: A⃗ + (-A⃗) = 0⃗.
How is vector addition used in physics?
Vector addition is used to combine forces, velocities, accelerations, and other vector quantities. The resultant represents the net effect of all vectors.