Calculator Inputs
Enter components for up to five vectors. Blank vector rows are ignored.
Example Data Table
| Vector | Multiplier | X | Y | Z | Scaled Vector |
|---|---|---|---|---|---|
| A | 1 | 3 | 4 | 1 | <3, 4, 1> |
| B | 1 | -2 | 5 | 2 | <-2, 5, 2> |
| Resultant | — | 1 | 9 | 3 | <1, 9, 3> |
Formula Used
Each vector is written as v = <x, y, z>. The calculator can also multiply each vector by a scalar k. The resultant vector is found by adding matching components.
R = Σ(kv) = <Σ(kx), Σ(ky), Σ(kz)>
Magnitude is calculated as |R| = √(Rx² + Ry² + Rz²). The unit vector is R divided by |R|. The equilibrant is -R.
For the planar angle, the calculator uses θ = atan2(Ry, Rx). For three-dimensional direction angles, it uses α = cos⁻¹(Rx / |R|), β = cos⁻¹(Ry / |R|), and γ = cos⁻¹(Rz / |R|).
How to Use This Calculator
- Select 2D or 3D mode.
- Enter each vector name, multiplier, and components.
- Leave unused vector rows blank.
- Choose decimal places for the final output.
- Press the calculate button.
- Review the result above the form.
- Download the CSV or PDF report when needed.
Vector Addition Guide
What This Tool Does
This vector add calculator helps you combine several vectors in a clean and direct way. It works with two-dimensional and three-dimensional vectors. You can enter horizontal, vertical, and depth components. You may also apply a scalar multiplier to each vector. This is helpful when a vector is repeated, reversed, resized, or weighted before addition.
Why Components Matter
Vector addition is easiest when every vector is split into components. The calculator adds all x components together. It also adds all y components together. In three-dimensional mode, it adds all z components too. The final answer is the resultant vector. This vector shows the total direction and strength of all entered vectors.
Advanced Output Values
The calculator does more than add components. It also finds magnitude. Magnitude tells how long or strong the resultant vector is. The planar angle shows direction from the positive x axis. The unit vector shows direction with length one. The equilibrant shows the opposite vector. It can balance the resultant.
Practical Uses
This tool is useful for physics, mechanics, navigation, engineering, graphics, and classroom work. Forces can be combined. Velocities can be added. Displacements can be checked. Game movement vectors can be tested. Survey data and motion paths can also be reviewed. The multiplier option makes repeated operations faster.
Reading the Result
A positive component points along the positive axis. A negative component points in the opposite direction. A zero component means no movement along that axis. The magnitude is always nonnegative. Direction angles help describe the vector position in space. The CSV option is best for spreadsheet review. The PDF option is best for simple reports.
Accuracy Tips
Use the same units for every vector. Do not mix meters with feet unless values are converted first. Enter signs carefully. A negative sign changes direction. Increase decimal places when small differences matter. Check the example table before entering larger problems. Clear inputs make cleaner reports and reduce mistakes.
FAQs
What is vector addition?
Vector addition combines matching components. All x values are added together. All y values are added together. In 3D, all z values are also added.
Can this calculator handle 3D vectors?
Yes. Select 3D mode, then enter x, y, and z components for each vector. The calculator returns resultant components and direction angles.
What does the multiplier field do?
The multiplier scales a vector before addition. A value of 2 doubles it. A value of -1 reverses it before adding.
What is the magnitude of a vector?
Magnitude is the length of the vector. It is found from the square root of the sum of squared resultant components.
What is a unit vector?
A unit vector has length one. It keeps the same direction as the resultant vector but removes the size effect.
What is an equilibrant vector?
The equilibrant is the opposite of the resultant vector. When added to the resultant, the final sum becomes zero.
Can I export the answer?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a simple report.
Should all vectors use the same unit?
Yes. Use the same unit before adding vectors. Convert values first if your data uses mixed units.