Compute dot and cross products from vectors. Review angles, projections, determinants, and matrix interpretation instantly. Build intuition for magnitude, direction, area, and orthogonality fast.
| Vector A | Vector B | Dot Product | Cross Product | Angle |
|---|---|---|---|---|
| [2, 1] | [1, 3] | 5 | 5 | 45.0000° |
| [1, 2, 3] | [4, 5, 6] | 32 | [-3, 6, -3] | 12.9332° |
| [3, 0, 0] | [0, 4, 0] | 0 | [0, 0, 12] | 90.0000° |
Dot product: A · B = axbx + ayby + azbz
2D cross scalar: A × B = axby − aybx
3D cross vector: A × B = [aybz − azby, azbx − axbz, axby − aybx]
Magnitude: |A| = √(ax² + ay² + az²)
Angle: cos(θ) = (A · B) / (|A||B|)
Projection of A on B: projB(A) = (A · B / |B|²)B
Area from cross product: |A × B| gives parallelogram area. Half of it gives triangle area.
Choose 2D or 3D first. Enter the components of vector A and vector B. Select whether you want a full comparison, only the dot product, or only the cross product.
Set the decimal places you want. Pick degrees or radians for the angle output. Use the normalization option when you want unit vectors before the main calculation.
Click the calculate button. The result appears above the form. You can then download the result as a CSV file or a PDF file for reporting or classroom use.
Dot product and cross product describe two different vector ideas. The dot product measures alignment. The cross product measures turning strength and perpendicular direction. Students often mix them. This calculator separates both results in one place. It helps you compare size, direction, projection, and angle without switching formulas.
The dot product fits naturally into matrix notation. You can write one vector as a row matrix and the other as a column matrix. Multiplying them produces one scalar value. That scalar tells you how much one vector points along another. A positive result shows similar direction. A negative result shows opposite direction. A zero result shows orthogonality.
The cross product behaves differently. In 3D, it returns a new vector that is perpendicular to both inputs. Its magnitude equals the area of the parallelogram made by the two vectors. In 2D, people often use the scalar determinant form instead. That scalar still captures orientation and signed area. This makes the cross product useful in geometry, physics, engineering, and graphics.
This calculator goes beyond one number. It computes magnitudes, angle, cosine similarity, projections, area, and vector relationship. It also lets you switch between 2D and 3D cases. That is useful because the cross product is commonly taught in three dimensions, while dot product works in any matching dimension.
Use the dot product when you need similarity, projection, work, or angle. Use the cross product when you need a perpendicular direction, torque style reasoning, orientation, or surface area. When both are shown together, the meaning becomes clearer. The dot product answers how much two vectors agree. The cross product answers how strongly they rotate away from each other. Seeing both side by side builds better vector intuition.
The dot product tells you how strongly two vectors align. A positive value means similar direction. Zero means perpendicular. A negative value means opposite direction.
The cross product gives a perpendicular result in 3D. Its magnitude shows area. In 2D, the related scalar determinant shows orientation and signed area.
Yes. This calculator returns the 2D scalar form. It is based on the determinant axby − aybx and is useful for orientation checks.
Parallel vectors make no turning area. Their sine term becomes zero, so the cross product magnitude becomes zero as well.
A negative dot product means the angle between vectors is greater than 90 degrees. The vectors point generally in opposite directions.
Normalization changes each vector into a unit vector. This removes scale and keeps direction. It is useful when you want shape comparison instead of raw size.
It uses vectors directly, but it also explains the dot product through matrix notation. That makes the scalar result easier to connect with linear algebra.
Choose dot for angle, similarity, and projection. Choose cross for perpendicular direction, area, and orientation. Each operation answers a different geometric question.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.