Dot Product: worked example
| Vector A: x | 1 |
|---|---|
| Vector A: y | 2 |
| Vector A: z | 3 |
| Vector B: x | 4 |
| Vector B: y | 5 |
| Vector B: z | 6 |
| Dot product | 32 |
| Angle between vectors (degrees) | 12.9 |
| Cross product magnitude | 7.3 |
Dot Product: formula
Enter the x, y and z components of each vector. For 2D vectors set z to zero. The dot product multiplies matching components and adds them, giving a single number.
The calculator also finds the angle between the vectors from the dot product and their lengths, and the magnitude of the cross product. Use consistent units for all components.
Dot Product: changing Vector A: x
| Vector A: x | Dot product | Angle between vectors (degrees) | Cross product magnitude |
|---|---|---|---|
| -1 | 24 | 43 | 22.4 |
| 0 | 28 | 27.7 | 14.7 |
| 1 (example) | 32 | 12.9 | 7.3 |
| 2 | 36 | 5.7 | 3.6 |
| 3 | 40 | 13.6 | 9.7 |
Dot Product: compare input ranges
| Input | Tested from | To | Dot product: low input | Dot product: high input | Swing |
|---|---|---|---|---|---|
| Vector A: z | 2 | 4 | 26 | 38 | 12 (38%) |
| Vector B: z | 4 | 8 | 26 | 38 | 12 (38%) |
| Vector A: y | 1 | 3 | 27 | 37 | 10 (31%) |
| Vector A: x | 0 | 2 | 28 | 36 | 8 (25%) |
| Vector B: y | 4 | 6 | 30 | 34 | 4 (13%) |
| Vector B: x | 3 | 5 | 31 | 33 | 2 (6.3%) |
Dot Product: individual input comparisons
| Input | Value | Dot product | Angle between vectors (degrees) | Cross product magnitude |
|---|---|---|---|---|
| Vector A: x | -1 | 24 | 43 | 22.4 |
| 0 | 28 | 27.7 | 14.7 | |
| 1 | 32 | 12.9 | 7.3 | |
| 2 | 36 | 5.7 | 3.6 | |
| 3 | 40 | 13.6 | 9.7 | |
| Vector A: y | 0 | 22 | 37.5 | 16.9 |
| 1 | 27 | 21.9 | 10.9 | |
| 2 | 32 | 12.9 | 7.3 | |
| 3 | 37 | 14.7 | 9.7 | |
| 4 | 42 | 20.2 | 15.4 | |
| Vector A: z | 1 | 20 | 21.5 | 7.9 |
| 2 | 26 | 9 | 4.1 | |
| 3 | 32 | 12.9 | 7.3 | |
| 4 | 38 | 19.1 | 13.2 | |
| 5 | 44 | 23.7 | 19.3 | |
| Vector B: x | 2 | 30 | 6 | 3.2 |
| 3 | 31 | 8 | 4.4 | |
| 4 | 32 | 12.9 | 7.3 | |
| 5 | 33 | 18 | 10.7 | |
| 6 | 34 | 22.7 | 14.2 | |
| Vector B: y | 3 | 28 | 16.6 | 8.4 |
| 4 | 30 | 13.5 | 7.2 | |
| 5 | 32 | 12.9 | 7.3 | |
| 6 | 34 | 14.4 | 8.7 | |
| 7 | 36 | 16.8 | 10.9 | |
| Vector B: z | 2 | 20 | 37.2 | 15.2 |
| 4 | 26 | 23 | 11 | |
| 6 | 32 | 12.9 | 7.3 | |
| 8 | 38 | 7.6 | 5.1 | |
| 10 | 44 | 8 | 6.2 |
Dot Product: combined input changes
| Vector A: x / Vector A: y | 0 | 2 | 4 |
|---|---|---|---|
| -1 | 14 | 24 | 34 |
| 0 | 18 | 28 | 38 |
| 1 | 22 | 32 | 42 |
| 2 | 26 | 36 | 46 |
| 3 | 30 | 40 | 50 |
Dot Product: which assumptions move the result?
Dot Product: values and definitions
| Quantity and definition | Value used |
|---|---|
| Vector A: x | 1 |
| Vector A: y | 2 |
| Vector A: z | 3 |
| Vector B: x | 4 |
| Vector B: y | 5 |
| Vector B: z | 6 |
| Dot product | 32 |
| Angle between vectors (degrees) | 12.9 |
| Cross product magnitude | 7.3 |
Dot Product: half and double Vector A: z
| Vector A: z | Dot product |
|---|---|
| 1.5 | 23 |
| 3 | 32 |
| 6 | 50 |
Common error: Reading a large dot product as a large angle. It is the other way around: for vectors of the same length, the more they line up, the larger the value and the smaller the angle between them.
The dot product is a number, while the cross product is a vector defined only in three dimensions and depends on the order of the vectors. The angle result is unreliable when either vector is very small. Rounding affects nearly parallel vectors.
Vectors must have the same dimension.
Algebra & Arithmetic calculators: Mixed Number to Improper Fraction Calculator, Exponential Growth Calculator, Quadratic Formula Calculator, Fraction and Ratio Simplifier Calculator, Exponent Calculator, Root Calculator.
The dot product multiplies matching parts of two vectors and adds them up, and the result says how much they point the same way. The cross product magnitude, 7.35 here, measures the area of the parallelogram they span.
Enter the vectors in matching order: x, y and z from the same coordinate system for both. A mismatch changes the angle and the dot product.
See lengths and distances on the distance and slope calculator, angles on the trigonometry page, and matrix determinants on the 3x3 determinant calculator, which relates to the cross product. The physics calculators use vectors for force and work. The midpoint page handles coordinates.
What does the cross-product magnitude represent?
The area of the parallelogram spanned by the two 3D vectors. For (1,2,3) and (4,5,6), it is √54, or about 7.34847.
What does a dot product of zero mean?
For two nonzero vectors, a zero dot product means a right angle between them in this Euclidean model. The zero vector also has zero dot product with every vector, but has no defined direction. Check both lengths before using the result as an angle or perpendicular-direction test.
How do I find the angle between two vectors?
Divide their dot product by the product of their lengths and take the inverse cosine of the result.
How is the cross product different?
It produces a vector perpendicular to both inputs, with a length equal to the area of the parallelogram they form.