Calculate the resultant vector from up to three 2D or 3D vectors and find its X, Y, Z components, magnitude, direction angle, and elevation.
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Vector Addition Formula
Vector addition is done by adding matching components. Add the x components together, add the y components together, and add the z components together if you are working in 3D.
- A, B, C are the input vectors. The third vector is optional.
- Ax, Bx, Cx are the x components of the input vectors.
- Ay, By, Cy are the y components of the input vectors.
- Az, Bz, Cz are the z components of the input vectors.
- Rx, Ry, Rz are the components of the resultant vector.
- |R| is the magnitude, or length, of the resultant vector.
- θ is the direction angle in the xy plane, measured in degrees from the positive x-axis.
- φ is the elevation angle from the xy plane. It is only used when a z component is entered.
For 2D vector addition, leave the z fields blank. The z components are treated as 0, so the magnitude becomes √(Rx² + Ry²).
For 3D vector addition, enter z components where needed. The calculator adds the z components and uses them in the magnitude. It also shows the elevation angle when any z value is entered.
The direction angle uses atan2, so it handles all four quadrants correctly. If the resultant has x = 0 and y = 0, the xy direction is undefined.
Common Vector Addition Results
These tables can help you check the meaning of the result.
| Result components | Direction in xy plane | What it means |
|---|---|---|
| Rx > 0, Ry = 0 | 0° | Points along the positive x-axis. |
| Rx = 0, Ry > 0 | 90° | Points along the positive y-axis. |
| Rx < 0, Ry = 0 | 180° | Points along the negative x-axis. |
| Rx = 0, Ry < 0 | 270° | Points along the negative y-axis. |
| Rx = 0, Ry = 0 | Undefined | There is no direction in the xy plane. |
| Input type | Use these fields | Result shown |
|---|---|---|
| 2D vectors | x and y only | Resultant x, resultant y, magnitude, and xy direction angle. |
| 3D vectors | x, y, and z | Resultant x, y, z, 3D magnitude, xy direction angle, and elevation angle. |
| Two vectors | Vector 1 and Vector 2 | The sum of the first two vectors. |
| Three vectors | Vector 1, Vector 2, and Vector 3 | The sum of all three vectors. |
Vector Addition Examples
Example 1: Add two 2D vectors
Add A = (3, 4) and B = (2, -1).
Add the x components:
Add the y components:
The resultant vector is:
The magnitude is:
The direction angle is:
Example 2: Add three 3D vectors
Add A = (1, 2, 3), B = (4, -2, 1), and C = (-3, 5, 2).
Add each component:
The resultant vector is:
The magnitude is:
The xy direction angle is:
The elevation angle is:
FAQ
How do you add vectors with components?
Add the matching components. The x values add together, the y values add together, and the z values add together if the vectors are in 3D. For example, (2, 5) + (4, -1) = (6, 4).
What does the magnitude of the resultant vector mean?
The magnitude is the length of the resultant vector. In physics, it can represent the size of a total force, displacement, velocity, or other vector quantity. The direction tells where that total vector points.
Why is the direction angle sometimes undefined?
The direction angle in the xy plane is undefined when the resultant x component and y component are both 0. In that case, the vector has no direction in the xy plane. If the full 3D vector is also (0, 0, 0), the whole vector has no direction.
