Калькулятор Скалярного Произведения

Рассчитайте скалярное произведение двух векторов в 2D, 3D или более высоких измерениях. Узнайте угол между векторами, их длины, скалярные и векторные проекции, геометрическую интерпретацию и пошаговые формулы с интерактивной векторной диаграммой.

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О Калькулятор Скалярного Произведения

The Dot Product Calculator computes the scalar product of two vectors in 2D, 3D, or higher dimensions using the algebraic formula \(\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i\). Enter the components of your two vectors to instantly get the dot product, angle between vectors, magnitudes, scalar and vector projections, geometric interpretation, and a step-by-step solution with an interactive vector diagram.

Real-World Applications

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Physics: Work

W = F · d calculates work done by a force

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Lighting

Lambertian shading uses N · L for brightness

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Machine Learning

Cosine similarity measures document relevance

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Game Dev

Collision detection and field-of-view checks

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Signal Processing

Correlation between signals via inner product

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Engineering

Resolving force components along directions

Key Formulas

Property	Formula	Description
Dot Product	\(\vec{a} \cdot \vec{b} = \sum a_i b_i\)	Sum of component-wise products
Geometric Form	\(\vec{a} \cdot \vec{b} = \|\vec{a}\|\|\vec{b}\|\cos\theta\)	Product of magnitudes times cosine of angle
Angle	\(\theta = \arccos\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\|\vec{b}\|}\)	Angle between the two vectors (0° to 180°)
Magnitude	\(\|\vec{a}\| = \sqrt{\sum a_i^2}\)	Length (Euclidean norm) of a vector
Scalar Projection	\(\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|}\)	Signed length of a's shadow on b
Vector Projection	\(\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2}\vec{b}\)	Vector component of a along b

Dot Product vs. Cross Product

Dot Product (a · b)

Produces a scalar value. Works in any dimension (2D, 3D, nD). Measures how much two vectors point in the same direction. Zero when vectors are perpendicular. Used for projections, angles, and work calculations.

Cross Product (a × b)

Produces a vector perpendicular to both inputs. Only defined in 3D (and 7D). Magnitude equals the area of the parallelogram formed by the vectors. Zero when vectors are parallel. Used for torque, normals, and area calculations.

Understanding the Geometric Interpretation

The dot product has a deep geometric meaning: \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\). This tells us:

Positive dot product (θ < 90°): vectors point in a generally similar direction.
Zero dot product (θ = 90°): vectors are perpendicular (orthogonal) — this is the foundation of orthogonality tests in linear algebra.
Negative dot product (θ > 90°): vectors point in generally opposite directions.

The scalar projection of \(\vec{a}\) onto \(\vec{b}\) gives the signed length of \(\vec{a}\)'s "shadow" when light shines perpendicular to \(\vec{b}\). The vector projection gives this shadow as an actual vector along \(\vec{b}\).

How to Use the Dot Product Calculator

Select the dimension: Choose 2D, 3D, 4D, or Custom for higher dimensions. Click a quick example to auto-fill sample values.
Enter Vector a: Type the components separated by commas (e.g., 3, 4, 5 for a 3D vector).
Enter Vector b: Type the components of the second vector in the same dimension.
Watch the live preview: The vector diagram updates in real-time as you type, showing the spatial relationship and angle between vectors.
Click Calculate: Press the button to get the full results including dot product, angle, magnitudes, projections, interpretation, and step-by-step formulas.

Properties of the Dot Product

Commutative: \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)
Distributive: \(\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}\)
Scalar multiplication: \((k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})\)
Self dot product: \(\vec{a} \cdot \vec{a} = |\vec{a}|^2\) (square of the magnitude)
Cauchy-Schwarz inequality: \(|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|\)

FAQ

What is the dot product of two vectors?

The dot product (also called scalar product or inner product) of two vectors a and b is the sum of the products of their corresponding components: a · b = a₁b₁ + a₂b₂ + ... + aₙbₙ. It produces a single scalar value, not a vector. Geometrically, it equals |a||b|cos θ, where θ is the angle between the vectors.

How do you find the angle between two vectors using the dot product?

Use the formula θ = arccos(a · b / (|a| × |b|)). First compute the dot product, then divide by the product of the two magnitudes to get cos θ, and finally take the inverse cosine. The result is always between 0° and 180°.

What does it mean when the dot product is zero?

When the dot product of two non-zero vectors is zero, the vectors are perpendicular (orthogonal), meaning they form a 90° angle. This is one of the most important properties of the dot product and is widely used in linear algebra, physics, and computer graphics to test orthogonality.

What is the difference between dot product and cross product?

The dot product produces a scalar and measures how much two vectors align. It works in any dimension. The cross product produces a vector perpendicular to both inputs and is only defined in 3D. The dot product is used for angles and projections; the cross product is used for torque, surface normals, and area calculations.

What is vector projection?

The vector projection of a onto b is the component of vector a that lies along the direction of b. It equals (a · b / |b|²) × b. The scalar projection (also called the component) is a · b / |b|, giving the signed length of this projection. If the scalar projection is positive, a has a component in the direction of b; if negative, it points opposite to b.

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by miniwebtool team. Updated: 2026-04-09

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