Free Dot Product Calculator | Vector Dot Product Calculator
How It Works
Enter Vectors
Input two vectors with same dimensions
Calculate Product
Multiply corresponding elements and sum
What is Dot Product Calculator?
What
A dot product calculator multiplies two vectors to get a single number. It's like combining two lists of numbers in a special mathematical way.
Why
Used in physics, engineering, computer graphics, and machine learning to calculate angles, projections, and similarities between vectors.
Applications
Game development, 3D graphics, machine learning algorithms, physics simulations, and engineering calculations.
Common Examples
Example 1: 2D Vectors
Vectors: [3, 4] and [2, 1]
Calculation: (3 × 2) + (4 × 1) = 6 + 4 = 10
Result: 10
Example 2: 3D Vectors
Vectors: [1, 2, 3] and [4, 5, 6]
Calculation: (1 × 4) + (2 × 5) + (3 × 6) = 4 + 10 + 18 = 32
Result: 32
Example 3: Perpendicular Vectors
Vectors: [1, 0] and [0, 1]
Calculation: (1 × 0) + (0 × 1) = 0 + 0 = 0
Result: 0 (perpendicular)
Example 4: Negative Values
Vectors: [2, -3] and [-1, 4]
Calculation: (2 × -1) + (-3 × 4) = -2 + (-12) = -14
Result: -14
Calculation Table
| Vector A | Vector B | Dot Product | Calculation |
|---|---|---|---|
| [1, 0] | [0, 1] | 0 | (1×0) + (0×1) = 0 |
| [2, 3] | [4, 1] | 11 | (2×4) + (3×1) = 11 |
| [1, 2, 3] | [4, 5, 6] | 32 | (1×4) + (2×5) + (3×6) = 32 |
| [5, -2] | [3, 4] | 7 | (5×3) + (-2×4) = 7 |
| [1, 1, 1] | [2, 2, 2] | 6 | (1×2) + (1×2) + (1×2) = 6 |
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Real-World Applications
🎮 Video Games
Calculate lighting effects, collision detection, and camera angles in 3D games.
🤖 Machine Learning
Measure similarity between data points and calculate neural network weights.
🏗️ Engineering
Calculate forces, moments, and structural analysis in civil and mechanical engineering.
📱 Computer Graphics
Render 3D objects, calculate shadows, and create realistic visual effects.
🔬 Physics
Calculate work done by forces, electromagnetic fields, and quantum mechanics.
📊 Data Analysis
Find correlations between datasets and perform statistical calculations.
Dot Product Formula and Method
Where a and b are vectors with n components each
Step-by-Step Method:
- Check that both vectors have the same number of components
- Multiply the first components: a₁ × b₁
- Multiply the second components: a₂ × b₂
- Continue for all components
- Add all the products together
Properties:
- Result is always a single number (scalar)
- Commutative: a · b = b · a
- If result is 0, vectors are perpendicular
- If result is positive, vectors point in similar directions
- If result is negative, vectors point in opposite directions
Frequently Asked Questions
What is a dot product in simple terms?
A dot product is a way to multiply two lists of numbers (vectors) to get one single number. You multiply matching positions and add them all up.
Why do I sometimes get zero as the result?
A zero result means the vectors are perpendicular (at 90 degrees to each other). This is mathematically significant and useful in many applications.
Can vectors have different lengths?
No, both vectors must have the same number of components. You cannot calculate the dot product of [1,2] and [3,4,5] because they have different dimensions.
What does a negative dot product mean?
A negative dot product means the vectors point in generally opposite directions (the angle between them is greater than 90 degrees).
How is dot product used in real life?
Dot product is used in computer graphics for lighting calculations, in physics for work and energy, in machine learning for similarity measures, and in engineering for force calculations.
Is dot product the same as cross product?
No, they are different. Dot product gives you a single number, while cross product gives you another vector. Dot product works in any dimension, cross product is mainly for 3D vectors.