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Vector Addition Calculator

Add two vectors component-wise in 2D or 3D space.

Reviewed by Christopher FloiedPublished Updated

This free online vector addition calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Set to 0 for 2D

Set to 0 for 2D

Results

Sum x

5

Sum y

7

Sum z

9

Magnitude of Result

12.4499

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Vector Addition Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Vector Addition Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Vector Addition Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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Associative Property Calculator

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About Vector Addition Calculator

Vector addition combines two vectors component-wise to produce a resultant vector. Geometrically, it corresponds to placing the tail of the second vector at the tip of the first (the 'tip-to-tail' method), and the resultant goes from the tail of the first to the tip of the second. This is equivalent to the parallelogram rule: the resultant is the diagonal of the parallelogram formed by the two vectors. Vector addition is fundamental to physics (combining forces, velocities, displacements), engineering (structural analysis, signal processing), and computer graphics (translations, animations). This calculator adds two vectors in 2D or 3D, providing the resultant vector and its magnitude.

The Math Behind It

Vector addition in component form is straightforward: given A = (x1, y1, z1) and B = (x2, y2, z2), the sum A + B = (x1 + x2, y1 + y2, z1 + z2). This operation is commutative (A + B = B + A), associative ((A + B) + C = A + (B + C)), and has the zero vector as its identity element. The negative of a vector A is -A = (-x1, -y1, -z1), and subtraction is defined as A - B = A + (-B). The triangle inequality states |A + B| <= |A| + |B|, with equality only when A and B point in the same direction. In physics, the net force on an object is the vector sum of all individual forces. The principle of superposition states that effects from multiple sources add as vectors. In navigation, displacement vectors add to give the total displacement. Vector addition extends naturally to any number of dimensions.

Formula Reference

Vector Addition

A + B = (x₁ + x₂, y₁ + y₂, z₁ + z₂)

Variables: A = (x₁, y₁, z₁), B = (x₂, y₂, z₂)

Worked Examples

Example 1: Adding two 3D vectors

Add A = (1, 2, 3) and B = (4, 5, 6)

Step 1:x: 1 + 4 = 5
Step 2:y: 2 + 5 = 7
Step 3:z: 3 + 6 = 9
Step 4:Result: (5, 7, 9)
Step 5:Magnitude: √(25 + 49 + 81) = √155 ≈ 12.45

A + B = (5, 7, 9), magnitude ≈ 12.45

Common Mistakes & Tips

  • !Multiplying components instead of adding them.
  • !Adding magnitudes directly instead of adding components first (magnitudes don't add unless vectors are parallel).
  • !Confusing vector addition with the dot product or cross product.
  • !Forgetting to include all components (e.g., omitting the z component in 3D).

Related Concepts

Vector Magnitude

Find the length of the resultant vector.

Dot Product

Measure the alignment between vectors.

Unit Vector

Normalize the result to get its direction.

Frequently Asked Questions

Is the magnitude of the sum equal to the sum of magnitudes?

Only if the vectors point in the same direction. In general, |A + B| <= |A| + |B| (triangle inequality).

Can I subtract vectors?

Yes. Subtraction is adding the negative: A - B = A + (-B), where -B reverses the direction.

What is the zero vector?

The zero vector (0, 0, 0) is the identity element for addition: A + 0 = A.

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