32
3.7417
8.775
0.974632
0.225726
rad
12.9332
°
32
3.7417
8.775
0.974632
0.225726
rad
12.9332
°
The Dot Product Calculator computes the scalar (inner) product of two three-dimensional vectors and determines the angle between them. The dot product is one of the two fundamental multiplication operations for vectors — the other being the cross product — and it plays a central role in geometry, physics, and data science.
Formally, the dot product of vectors A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂) is defined algebraically as A · B = x₁x₂ + y₁y₂ + z₁z₂. The result is a scalar, not a vector. Geometrically, this equals the product of the two magnitudes times the cosine of the angle between them: A · B = |A| |B| cos θ. This dual interpretation — algebraic and geometric — is what makes the dot product so powerful and versatile.
The sign of the dot product immediately reveals the angular relationship between two vectors. A positive dot product means the vectors point in roughly the same direction (acute angle, θ < 90°). A negative dot product means they point in roughly opposite directions (obtuse angle, θ > 90°). A dot product of exactly zero means the vectors are orthogonal (perpendicular), one of the most important geometric relationships in mathematics.
In physics, the dot product computes work done by a force: W = F · d, where F is the force vector and d is the displacement vector. Only the component of force parallel to the displacement does work, and the dot product elegantly captures this projection. Similarly, power is the dot product of force and velocity, and electric flux involves the dot product of the electric field with an area vector.
In computer graphics, the dot product determines how brightly a surface is illuminated. The Lambertian lighting model computes brightness as the dot product of the surface normal and the light direction vector. This single calculation drives the shading of virtually every 3D-rendered image. In machine learning and information retrieval, the dot product (and its normalized form, cosine similarity) measures how similar two feature vectors or document embeddings are.
This calculator accepts two 3D vectors and computes their dot product, the angle between them in both degrees and radians, and the individual magnitudes. It serves students, engineers, physicists, and programmers who need quick, reliable vector computations without manual arithmetic.
The dot product is computed using the algebraic definition:
$$\mathbf{A} \cdot \mathbf{B} = x_1 x_2 + y_1 y_2 + z_1 z_2$$
The magnitudes of each vector are:
$$|\mathbf{A}| = \sqrt{x_1^2 + y_1^2 + z_1^2}, \quad |\mathbf{B}| = \sqrt{x_2^2 + y_2^2 + z_2^2}$$
The angle between the vectors is derived from the geometric form of the dot product:
$$\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}|\,|\mathbf{B}|}$$
$$\theta = \arccos\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}|\,|\mathbf{B}|}\right)$$
The cosine value is clamped to the interval [−1, 1] before applying arccos to prevent numerical errors from floating-point arithmetic. The angle is returned in both radians and degrees (multiplied by 180/π).
The dot product value tells you the combined magnitude of the parallel components. A large positive value means the vectors are closely aligned; a large negative value means they point in opposite directions. Zero means perpendicular. The angle ranges from 0° (parallel, same direction) to 180° (anti-parallel, opposite direction). An angle of 90° confirms orthogonality. The individual magnitudes are provided for reference and verification.
Inputs
Results
A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32. cos θ = 32 / (3.7417 × 8.7750) ≈ 0.9746. θ = arccos(0.9746) ≈ 12.93°. The vectors are nearly aligned.
Inputs
Results
A · B = (1)(0) + (0)(1) + (0)(0) = 0. The dot product is zero, confirming the standard basis vectors î and ĵ are orthogonal. θ = 90° exactly.
A dot product of zero means the two vectors are orthogonal (perpendicular). They have no component in common along any shared direction. This is one of the most important tests in linear algebra and geometry.
Yes. The dot product is negative when the angle between the vectors is greater than 90° (obtuse). This means the vectors point in generally opposite directions. The most negative value occurs at 180° (anti-parallel vectors).
The dot product produces a scalar and measures parallelism (how aligned two vectors are). The cross product produces a vector perpendicular to both inputs and measures perpendicularity (the area of the parallelogram they span). They are complementary operations.
The dot product computes work (W = F · d), power (P = F · v), electric flux (Φ = E · A), and magnetic flux. Any time you need the component of one vector along another, the dot product is the tool.
Yes. A · B = B · A always. The order of the vectors does not matter. This follows directly from the commutativity of multiplication of real numbers.
Cosine similarity is the dot product of two vectors divided by the product of their magnitudes: cos θ = (A · B) / (|A| |B|). It ranges from −1 to 1 and measures directional similarity regardless of magnitude. It is widely used in text analysis, recommendation systems, and machine learning.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Matrix Addition Calculator
Linear Algebra Calculators
Matrix Multiplication Calculator
Linear Algebra Calculators
Matrix Determinant Calculator
Linear Algebra Calculators
Matrix Inverse Calculator
Linear Algebra Calculators
Matrix Transpose Calculator
Linear Algebra Calculators
Matrix Rank Calculator
Linear Algebra Calculators
