Answer first
What this calculator tells you
Calculate the dot product, the angle and the cross product magnitude of two 3D vectors. Find the angle between two directions or test whether two vectors are perpendicular. Formula: A·B = x₁x₂ + y₁y₂ + z₁z₂; angle = arccos(A·B ÷ (|A| |B|)); |A × B| = |A| |B| sin θ. At the worked-example inputs, the dot product is 32. Holding every other input steady, moving vector a: x from -1 to 3 moves the result from 24 to 40.
Transparent method
The formula
Find the angle between two directions or test whether two vectors are perpendicular.
Worked example
Example inputs
How to interpret the result
The dot product multiplies matching parts of two vectors and adds them up, and the result says how much they point the same way. Vectors (1, 2, 3) and (4, 5, 6) give 32, and dividing by their lengths gives an angle of about 12.9 degrees. A dot product of zero means the vectors are perpendicular. The cross product magnitude, 7.35 here, measures the area of the parallelogram they span.
At the worked-example inputs the dot product is 32. It rises with vector a: z, vector b: z, vector a: y, vector a: x, vector b: y and vector b: x.
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects.
Before you rely on it
What to check
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.
The common error
Where people go wrong with dot product calculator
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.
Sensitivity evidence
How vector a: x changes the dot product
Holding every other input at the worked-example value, moving vector a: x from -1 to 3 moves the dot product from 24 to 40: a spread of 16, or 50% of the worked-example result.
| Vector A: x | Dot product | Angle between vectors (degrees) | Cross product magnitude |
|---|---|---|---|
| -1 | 24 | 43 | 22.4 |
| 0 | 28 | 27.7 | 14.7 |
| 1worked example | 32 | 12.9 | 7.3 |
| 2 | 36 | 5.7 | 3.6 |
| 3 | 40 | 13.6 | 9.7 |
Every input, tested
Which input moves the dot product most
Of the 6 inputs, vector a: z moves the dot product most (12 across the range tested) and vector b: x moves it least (2).
| Input | Tested from | To | Dot product at each end | Swing |
|---|---|---|---|---|
| Vector A: z | 2 | 4 | 26 to 38 | 12 (38%) |
| Vector B: z | 4 | 8 | 26 to 38 | 12 (38%) |
| Vector A: y | 1 | 3 | 27 to 37 | 10 (31%) |
| Vector A: x | 0 | 2 | 28 to 36 | 8 (25%) |
| Vector B: y | 4 | 6 | 30 to 34 | 4 (13%) |
| Vector B: x | 3 | 5 | 31 to 33 | 2 (6.3%) |
Two variables at once
Dot product by vector a: x and vector a: y
Across the grid the dot product runs from 14 to 50. Moving vector a: x from -1 to 3 shifts it by 16 at the middle column, and moving vector a: y from 0 to 4 shifts it by 20 at the middle row, so vector a: y is the bigger lever here.
| 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 |
The highlighted cell is the worked example: 32.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Vector A: x | 1 | Enter the vector a: x used in this calculation. |
| Vector A: y | 2 | Enter the vector a: y used in this calculation. |
| Vector A: z | 3 | Enter the vector a: z used in this calculation. |
| Vector B: x | 4 | Enter the vector b: x used in this calculation. |
| Vector B: y | 5 | Enter the vector b: y used in this calculation. |
| Vector B: z | 6 | Enter the vector b: z used in this calculation. |
| Dot product | 32 | |
| Angle between vectors (degrees) | 12.9 | |
| Cross product magnitude | 7.3 | |
Inputs, definitions and assumptions
Vector A: x
Enter the vector a: x used in this calculation. The prefilled worked-example value is 1.
Vector A: y
Enter the vector a: y used in this calculation. The prefilled worked-example value is 2.
Vector A: z
Enter the vector a: z used in this calculation. The prefilled worked-example value is 3.
Vector B: x
Enter the vector b: x used in this calculation. The prefilled worked-example value is 4.
Vector B: y
Enter the vector b: y used in this calculation. The prefilled worked-example value is 5.
Vector B: z
Enter the vector b: z used in this calculation. The prefilled worked-example value is 6.
How to use this calculator
- 1Verify the inputs. Gather vector a: x, vector a: y, vector a: z, vector b: x, vector b: y and vector b: z from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the dot product at 32. Store your own version of it as Scenario A.
- 3Test one change. Start with vector a: z, the input with the biggest effect here: moving vector a: z from 2 to 4 takes the dot product from 26 to 38, a swing of 38% of the worked-example figure.
- 4Check the extremes. At half the example vector a: z (1.5) the dot product is 23; at double (6) it is 50.
People also ask
Frequently asked questions
How do you calculate dot product?
A·B = x₁x₂ + y₁y₂ + z₁z₂; angle = arccos(A·B ÷ (|A| |B|)); |A × B| = |A| |B| sin θ. At the worked-example inputs the dot product is 32.
What does the dot product result mean?
Find the angle between two directions or test whether two vectors are perpendicular. At the worked-example inputs the dot product is 32. It rises with vector a: z, vector b: z, vector a: y, vector a: x, vector b: y and vector b: x.
How much does vector a: x change the dot product?
Holding every other input at the worked-example value, moving vector a: x from -1 to 3 moves the dot product from 24 to 40, a spread of 16.
What are the limits of this dot product calculator?
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects. The tables on this page test vector a: x only from -1 to 3; a value outside that range is not tabulated here.
Which input moves the dot product most in the dot product calculator?
Ranked by how far each moves the dot product across the range tested: vector a: z (12, 38%), vector b: z (12, 38%), vector a: y (10, 31%) and vector a: x (8, 25%).
If I double vector a: z in the dot product calculator, does the dot product double?
Doubling it from 3 to 6 takes the dot product from 32 to 50, which is 1.56 times the worked-example figure. So it grows, but by less than double. Halving it to 1.5 gives 23.
How much does vector a: y matter in the dot product calculator?
The worked example uses 2. Holding every other input at its worked-example value, moving vector a: y from 1 to 3 takes the dot product from 27 to 37, a swing of 31% of the worked-example figure.
How much does vector a: z matter in the dot product calculator?
The worked example uses 3. With the other inputs left at the worked example, moving vector a: z from 2 to 4 takes the dot product from 26 to 38, a swing of 38% of the worked-example figure.
How much does vector b: x matter in the dot product calculator?
The worked example uses 4. Holding every other input at its worked-example value, moving vector b: x from 3 to 5 takes the dot product from 31 to 33, a swing of 6.3% of the worked-example figure.
How much does vector b: y matter in the dot product calculator?
The worked example uses 5. With the other inputs left at the worked example, moving vector b: y from 4 to 6 takes the dot product from 30 to 34, a swing of 13% of the worked-example figure.
How much does vector b: z matter in the dot product calculator?
The worked example uses 6. Holding every other input at its worked-example value, moving vector b: z from 4 to 8 takes the dot product from 26 to 38, a swing of 38% of the worked-example figure.
Which inputs change the angle between vectors (degrees) in the dot product calculator?
At the worked-example inputs it is 12.9. Vector a: x takes it from 27.7 to 5.7, vector a: y takes it from 21.9 to 14.7, vector a: z takes it from 9 to 19.1 and vector b: x takes it from 8 to 18.
Which inputs change the cross product magnitude in the dot product calculator?
At the worked-example inputs it is 7.3. Vector a: x takes it from 14.7 to 3.6, vector a: y takes it from 10.9 to 9.7, vector a: z takes it from 4.1 to 13.2 and vector b: x takes it from 4.4 to 10.7.
What's the difference between a root and an exponent?
An exponent raises a value to a power (2³ = 8). A root reverses that operation, asking what value raised to a power gives the result (the cube root of 8 is 2). A root is the same as raising to a fractional exponent: the cube root of x equals x^(1/3).
Sources and evidence
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