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How to use
This vector calculator works with two 3D vectors A and B. It reports each magnitude, the dot product A·B, the cross product A×B and its magnitude, and the angle between the vectors when both are nonzero. A 2D vector can be represented by setting z=0.
- Enter the x, y and z components of A.
- Enter the components of B.
- Calculate the magnitudes, dot product, cross product and angle.
How it is calculated
Example
Example: A=(1,2,3), B=(4,5,6). A·B=32, A×B=(-3,6,-3), |A×B|=√54≈7.3485 and the angle is about 12.93°.
Important notes
If either vector is zero, its direction is undefined and so is the angle between the vectors; the calculator does not invent a zero-degree angle. The magnitude of A×B equals the area of the parallelogram spanned by A and B.
Worked examples and interpreting results
Vector A=(3,4,0) has length 5 and B=(0,0,2) has length 2. Their dot product is zero, so they are perpendicular even though neither vector is zero.
| Case | Calculation | Result |
|---|---|---|
| Length of vector A | √(3² + 4² + 0²) | 5 |
| Dot product | 3×0 + 4×0 + 0×2 | 0 |
| Angle between vectors | acos(0 ÷ (5×2)) | 90 ° |
How to check the result
The angle involving a zero vector is undefined because the denominator is zero. A dot product returns a scalar; a cross product in three dimensions returns a vector. The operations are not interchangeable.
Frequently asked questions
When are vectors perpendicular?
When A·B=0, provided neither vector is zero.
What does the cross product give?
A vector perpendicular to A and B following the right-hand rule.
Can I use 2D vectors?
Yes. Set both z components to zero.
What is the difference between dot and cross products?
The dot product returns a scalar measuring alignment and is zero for perpendicular vectors. The cross product returns a vector perpendicular to both and, in its familiar vector form, is defined in three dimensions.