Dot Product Calculator – Complete Guide to Vector Dot, Angle, Cosine Similarity and Projection
The Dot Product Calculator on MyTimeCalculator brings together all the standard vector operations built around the dot product. It works in both three dimensions and higher dimensions, making it useful for physics, engineering, computer graphics, data science and linear algebra.
For 3D work, you can enter components Ax, Ay, Az and Bx, By, Bz. For higher-dimensional work, you can switch to the n-D tab and enter vector components as lists. The calculator handles dot product, magnitudes, angles, cosine similarity, projection andated operations automatically.
1. Dot Product of Two Vectors
The dot product of two vectors A and B is defined algebraically and geometrically. For 3D vectors A = <Ax, Ay, Az> and B = <Bx, By, Bz>, the algebraic formula is:
Geometrically, the dot productates magnitudes and the angle between vectors:
where θ is the angle between A and B. Thisationship is fundamental in both 3D and n-dimensional spaces, and the calculator uses it to find angles and cosine similarity.
2. Magnitude of a Vector
The magnitude (or length) of a vector A is computed using the Euclidean norm. In 3D:
In n dimensions, the same pattern holds:
The calculator automatically computes |A| and |B| in everyevant tab, so you can analyze the strength of vectors as well as their direction.
3. Angle Between Vectors
Rewriting the geometric dot product formula, the cosine of the angle θ between two non-zero vectors is:
From here, the angle is:
The Dot Product Calculator uses thisationship and shows the angle in degrees, which is convenient for most physics and engineering problems. If one vector has zero magnitude, the angle is undefined and the calculator returns a clear message instead.
4. Cosine Similarity
Cosine similarity is a normalized version of the dot product that measures directional alignment:
The value ranges from −1 to 1:
- 1 indicates vectors pointing in exactly the same direction
- 0 indicates orthogonal (perpendicular) vectors
- −1 indicates vectors pointing in opposite directions
In the n-D tab, the calculator focuses on dot products and cosine similarity, which are widely used in data science and machine learning for comparing feature vectors and embeddings.
5. Projection of One Vector onto Another
Projection decomposes a vector into components parallel and perpendicular to another vector. For vectors A and B (with B ≠ 0), the scalar projection of A onto B is:
The vector projection of A onto B is:
Projection is especially useful when resolving forces along a direction, measuring how far a point moves in a given direction or extracting a component along a basis vector in linear algebra.
6. Cross Product and Area (3D)
While the dot product measures how aligned two vectors are, the cross product measures how perpendicular they are. For 3D vectors A and B:
The magnitude |A × B| equals the area of the parallelogram spanned by A and B. The calculator shows both the cross product vector and its magnitude, which is helpful for torque, rotational effects and surface area calculations.
7. n-Dimensional Dot Product and Angle
In n dimensions, vectors are written as ordered lists of components: A = (a1, a2, …, an), B = (b1, b2, …, bn).
The same cosine similarity and angle formulas apply, which makes the dot product central in high-dimensional geometry, optimization and machine learning. The n-D tab lets you experiment with such vectors by entering components separated by commas or spaces.
How to Use the Dot Product Calculator
- Select the appropriate tab: 3D calculations, projection, cross product or n-D mode.
- Enter vector components in the input fields or text boxes.
- Click the “Compute” button for the selected tab.
- Read the dot product, magnitudes, angles andated quantities in the result cards.
If any input is missing or not a valid number, the calculator displays an alert so you can correct the values. For n-D mode, both vectors must have the same number of components; otherwise, a helpful message appears.
Typical Use Cases
- Physics: work done by a force along a displacement, power, components along axes.
- Engineering: stress analysis, projections along beams, resolving vector loads.
- Computer graphics: lighting computations, shading, back-face culling, normals.
- Robotics: orientation checks, motion along specified directions.
- Data science and machine learning: cosine similarity between feature vectors and embeddings.
- Education: checking vector exercises, exploring geometric interpretations and building intuition.
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Dot Product Calculator FAQs
Frequently Asked Questions
Find quick answers to common questions dot product, cosine similarity, projections and angles between vectors.
The dot product is a scalar that measures how aligned two vectors are. It is computed as the sum of the products of corresponding components. In 3D, A · B = AxBx + AyBy + AzBz. A positive dot product indicates a generally similar direction, zero indicates perpendicular vectors and a negative dot product indicates opposite directions.
The calculator first computes the dot product and the magnitudes of each vector. It then uses the formula cos(θ) = (A · B) / (|A||B|) and applies the inverse cosine function to find θ in degrees. If either vector has zero magnitude, the angle is undefined and the calculator displays a message instead of a number.
Cosine similarity is the dot product divided by the product of the magnitudes: (A · B) / (|A||B|). It measures the directional similarity of two vectors regardless of their length. In the calculator, cosine similarity appears both in the 3D tab and in the n-D tab, where it is widely used to compare high-dimensional feature vectors and embeddings.
Yes. The calculator accepts any real numbers, including negative values and decimals, in both 3D and n-dimensional modes. All operations follow the standard algebraic formulas, and results are formatted with a configurable number of decimal places for readability.
In the n-D tab, you can type vector components separated by commas, spaces or a mix of both. For example, “1, 2, 3, 4” and “4 3 2 1” are both valid inputs. The calculator parses the lists, checks that both vectors have the same length, and then computes dot product, magnitudes, cosine similarity and the angle between them.
The dot product is defined only when vectors have the same number of components. If A has n components and B has m components with n ≠ m, there is no standard dot product. The calculator checks this to avoid misleading results and prompts you to correct the input dimensions when needed.
A dot product of zero means the vectors are orthogonal, or perpendicular. In practice, this might mean that a force is perpendicular to a displacement (so it does no work), or that two feature vectors in a data set share no linear correlation in the vector space model being used.
The dot product produces a scalar measuring alignment, while the cross product produces a 3D vector perpendicular to both inputs. The Dot Product Calculator focuses on dot-based operations and angle calculations, but it also includes a dedicated 3D cross product tab for convenience when you need area and perpendicular direction information.
Yes. Work in physics is the dot product of force and displacement. By entering the force and displacement vectors, you can compute the dot product and verify your calculations. The angle and cosine similarity also help you understand how much of the force is aligned with the direction of motion.
The calculator is designed for typical educational and practical use, so moderate vector sizes work well as long as they can be entered in the input field. For extremely high-dimensional data, it is usually more efficient toy on programming tools or numerical libraries, but this calculator is ideal for testing and understanding problems with a manageable number of components.