Calculator Inputs
Enter the x, y, and z components for both vectors. The calculator will find the angle, projections, and vector relationship.
Example Data Table
These examples help verify expected output and show common vector relationships.
| Vector A | Vector B | Dot Product | Approx. Angle | Relationship |
|---|---|---|---|---|
| (1, 2, 3) | (4, 5, 6) | 32 | 12.93° | Acute |
| (1, 0, 0) | (0, 1, 0) | 0 | 90.00° | Perpendicular |
| (1, 1, 0) | (-2, 1, 0) | -1 | 108.43° | Obtuse |
| (2, 4, 6) | (1, 2, 3) | 28 | 0.00° | Parallel |
Formula Used
1) Dot Product
For vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz):
A · B = (Ax × Bx) + (Ay × By) + (Az × Bz)
2) Magnitude of Each Vector
|A| = √(Ax² + Ay² + Az²)
|B| = √(Bx² + By² + Bz²)
3) Angle Between Two 3D Vectors
cos(θ) = (A · B) / (|A| × |B|)
θ = arccos((A · B) / (|A| × |B|))
4) Cross Product Magnitude
|A × B| = |A||B|sin(θ)
This helps confirm perpendicular and parallel relationships.
How to Use This Calculator
- Enter the x, y, and z components of Vector A.
- Enter the x, y, and z components of Vector B.
- Click Calculate Angle to process the vectors.
- Read the angle result shown above the form.
- Review dot product, projections, unit vectors, and relationship type.
- Use the 3D graph to visually compare both vectors.
- Download the result as CSV or PDF when needed.
Frequently Asked Questions
1) What formula is used to calculate the angle?
The calculator uses cos(θ) = (A·B) / (|A||B|). It then applies the inverse cosine function to get the angle in radians and converts it to degrees.
2) Why does a zero vector cause an error?
A zero vector has magnitude zero. Since the formula divides by both magnitudes, the angle becomes undefined when either vector is zero.
3) Can the angle between two vectors be greater than 180°?
No. The standard angle between two vectors is taken as the smaller angle, so it always lies between 0° and 180°.
4) Do negative vector components work correctly?
Yes. Negative values are normal in vector calculations. They affect direction and can produce acute, right, obtuse, parallel, or opposite-direction results.
5) How do I know if vectors are perpendicular?
If the dot product is zero, the vectors are perpendicular. In that case, the calculator returns an angle close to 90°.
6) How do I know if vectors are parallel?
Parallel vectors have a cross product magnitude near zero. If the dot product is positive, they point in the same direction. If negative, they point in opposite directions.
7) What is the difference between dot and cross product?
The dot product measures directional similarity and helps find the angle. The cross product creates a new vector perpendicular to both input vectors.
8) Can I use decimals in the vector components?
Yes. The calculator accepts integers and decimals, so it works for classroom examples, engineering values, and general 3D geometry problems.