Dot Product Calculator
Calculate the dot product (inner product) of two vectors with detailed step-by-step solutions. Understand the geometric interpretation, explore orthogonality, and visualize vector relationships in 2D and 3D space with our comprehensive dot product calculator.
Vector Input
Dot Product Theory
What is the Dot Product?
The dot product (also called inner product or scalar product) is a fundamental operation that takes two vectors and returns a scalar value. It measures how much two vectors point in the same direction and is essential in physics, engineering, and computer graphics.
Algebraic Definition:
A • B = a₁b₁ + a₂b₂ + a₃b₃ + ... + aₙbₙ
Geometric Definition:
A • B = |A| × |B| × cos(θ)
Where: |A| and |B| are magnitudes, θ is the angle between vectors
Calculation Steps
Step 1: Multiply Components
Multiply corresponding components: a₁×b₁, a₂×b₂, etc.
Step 2: Sum Products
Add all the products together to get the final result
Step 3: Interpret Result
Positive: acute angle, Zero: perpendicular, Negative: obtuse angle
Step 4: Find Angle (Optional)
Use θ = arccos(A•B / (|A|×|B|)) to find the angle
Key Properties
Commutative
A • B = B • A
Distributive
A • (B + C) = A • B + A • C
Scalar Multiplication
(kA) • B = k(A • B)
Self Dot Product
A • A = |A|²
Orthogonality
A • B = 0 ⟺ A ⊥ B
Cauchy-Schwarz
|A • B| ≤ |A| × |B|
Applications
Physics
Work calculation, force projections, energy computations
Computer Graphics
Lighting models, surface normals, view frustum culling
Machine Learning
Similarity measures, neural networks, feature comparisons
Signal Processing
Correlation analysis, filtering, pattern recognition
Geometry
Angle calculations, orthogonality testing, projections
Engineering
Structural analysis, optimization, control systems
Worked Examples
Example 1: 2D Dot Product
Problem:
Calculate A • B where A = [3, 4] and B = [2, 1]
Solution:
Step 1: Multiply components: (3)(2) + (4)(1)
Step 2: Calculate: 6 + 4 = 10
Step 3: Verify: |A| = 5, |B| = √5, cos(θ) = 10/(5√5) = 2/√5
Result: A • B = 10
Example 2: 3D Orthogonal Vectors
Problem:
Calculate A • B where A = [1, 2, 3] and B = [2, -1, 0]
Solution:
Step 1: Multiply components: (1)(2) + (2)(-1) + (3)(0)
Step 2: Calculate: 2 - 2 + 0 = 0
Step 3: Since A • B = 0, vectors are orthogonal (perpendicular)
Result: A • B = 0, θ = 90°
Example 3: Angle Between Vectors
Problem:
Find the angle between A = [1, 0, 0] and B = [1, 1, 0]
Solution:
Step 1: A • B = (1)(1) + (0)(1) + (0)(0) = 1
Step 2: |A| = 1, |B| = √2
Step 3: cos(θ) = 1/(1×√2) = 1/√2
Step 4: θ = arccos(1/√2) = 45°
Result: Angle = 45°
Frequently Asked Questions
Dot product returns a scalar and measures how much vectors point in the same direction. Cross product returns a vector perpendicular to both input vectors and only works in 3D (or 7D). Dot product: A•B = scalar, Cross product: A×B = vector.
The dot product is a specific case of the more general inner product. In Euclidean space, they're the same thing. The term "inner product" is used in more abstract mathematical contexts, while "dot product" is common in physics and engineering.
When A•B = 0, the vectors are orthogonal (perpendicular). This means they meet at a 90° angle. This is a crucial concept in many applications, from 3D graphics to machine learning feature selection.
Yes! A negative dot product means the vectors point in generally opposite directions (obtuse angle > 90°). Positive means they point in similar directions (acute angle < 90°). Zero means they're perpendicular.
Dot products are everywhere in ML! They're used in neural network computations, similarity measures between data points, attention mechanisms in transformers, and calculating distances in high-dimensional spaces. They're fundamental to linear algebra operations in ML.
Geometrically, A•B equals the magnitude of A times the magnitude of the projection of B onto A (or vice versa). It measures how much one vector extends in the direction of another. This is why it's related to the cosine of the angle between them.