Calculator Inputs
Example Data Table
| Vector A | Vector B | Direction of A | Angle Between | Use Case |
|---|---|---|---|---|
| (3, 4) | (1, 2) | 53.1301° | 10.3048° | Basic 2D direction |
| (-5, 6) | (4, -2) | 129.8056° | 143.9726° | Quadrant checking |
| (3, 4, 2) | (-2, 5, 1) | 53.1301° xy view | 60.3472° | 3D comparison |
Formula Used
The 2D direction angle uses theta = atan2(y, x). This gives the signed angle from the positive x-axis.
The angle is normalized with theta = theta mod 360 degrees. This keeps the result between 0 degrees and 360 degrees.
The magnitude formula is |v| = sqrt(x² + y² + z²). For 2D mode, z is treated as zero.
The angle between vectors uses cos(theta) = (A · B) / (|A||B|). The dot product is A · B = axbx + ayby + azbz.
Direction angles use alpha = arccos(x / |v|), beta = arccos(y / |v|), and gamma = arccos(z / |v|).
The unit vector is v / |v|. It shows direction without length.
How to Use This Calculator
Select 2D or 3D mode first. Choose the calculation type next. Enter the vector components for Vector A.
Enter Vector B when you need an included angle, dot product, cross product, or projection result.
Choose degrees, radians, or both. Set decimal precision for cleaner answers. Press the calculate button.
The result appears under the header and above the input form. Use the export buttons to save the report.
Vector Direction Angle Calculator Guide
Purpose
A vector direction angle explains where a vector points. It is not the same as vector length. Length shows size. Direction shows orientation. This calculator handles both ideas together. It works for simple 2D vectors and extended 3D vectors. You can study the angle from the positive x-axis. You can also compare two vectors with the dot product method.
Why Direction Matters
Direction angles are useful in geometry, physics, surveying, graphics, navigation, and engineering. A force vector may point upward. A velocity vector may point southeast. A displacement vector may move through space. The angle makes that direction readable. A normalized angle also avoids confusion from negative atan values.
Advanced Output
This tool gives more than one answer. It finds magnitude, bearing, quadrant, unit vector, direction angles, dot product, cross product, and projection. These values help you check work from several viewpoints. The bearing is useful for compass style direction. The quadrant label helps detect sign mistakes in x and y components.
2D and 3D Work
In 2D mode, the calculator studies x and y components. The z value is ignored. In 3D mode, the z component is included in magnitude, dot product, cross product, and axis direction angles. The alpha, beta, and gamma angles show how the vector leans relative to each coordinate axis.
Practical Notes
A zero vector has no direction. The calculator blocks that case because every direction would be equally possible. For angle comparison, both vectors must have length. Use more decimal places when checking homework or technical reports. Use fewer decimal places for quick layout planning. Export options help save results for notes, worksheets, and project records.
FAQs
1. What is a vector direction angle?
It is the angle showing where a vector points from a reference axis. In 2D, it is usually measured from the positive x-axis.
2. Why does the calculator use atan2?
atan2 uses both x and y signs. This places the vector in the correct quadrant and avoids common inverse tangent errors.
3. Can I use negative vector components?
Yes. Negative components are valid. They show direction toward the negative side of an axis and affect the quadrant.
4. What is the angle between two vectors?
It is the smaller included angle found from the dot product. It ranges from 0 degrees to 180 degrees.
5. What is a unit vector?
A unit vector has length one. It keeps the same direction as the original vector but removes the magnitude.
6. Why is the zero vector not allowed?
The zero vector has no clear direction. Its magnitude is zero, so direction formulas cannot divide by its length.
7. What does compass bearing mean?
Compass bearing measures clockwise from north. It is helpful for navigation style readings and map direction checks.
8. When should I choose 3D mode?
Choose 3D mode when your vector includes x, y, and z components. It adds gamma angle and full spatial products.