Advanced Vector Calculator Form
Example Data Table
| Vector A | Vector B | Operation | Expected Result |
|---|---|---|---|
| (3, 4, 2) | (1, 2, 5) | A + B | (4, 6, 7) |
| (3, 4, 2) | (1, 2, 5) | A · B | 21 |
| (3, 4, 2) | (1, 2, 5) | A × B | (16, -13, 2) |
| (3, 4, 2) | (1, 2, 5) | |A| | 5.3852 |
Formula Used
Addition: A + B = (Ax + Bx, Ay + By, Az + Bz)
Subtraction: A - B = (Ax - Bx, Ay - By, Az - Bz)
Magnitude: |A| = √(Ax² + Ay² + Az²)
Dot Product: A · B = AxBx + AyBy + AzBz
Cross Product: A × B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)
Angle: θ = cos⁻¹((A · B) / (|A||B|))
Unit Vector: Â = A / |A|
Projection: projB A = ((A · B) / |B|²)B
How to Use This Calculator
Choose a 2D or 3D vector mode. Enter the x, y, and z components for vector A and vector B. In 2D mode, the z values are treated as zero. Select a focused operation if you need one main answer. Select full report to see every supported result.
Enter a scalar value when you want to calculate kA. Choose the decimal precision for rounded answers. Add a unit label if your vectors describe force, motion, position, velocity, or distance. Press the calculate button. The result appears below the header and above the form. Use the CSV or PDF buttons to save the report.
Vector Calculator Guide
What This Tool Does
A vector describes size and direction together. It can model force, velocity, displacement, acceleration, and many other quantities. This calculator works with two-dimensional and three-dimensional vectors. It accepts component form, so each vector is entered through x, y, and z values. The tool then creates a complete report.
Why Components Matter
Components make vector work easier. A long arrow can be split into horizontal, vertical, and depth parts. Once the parts are known, addition and subtraction become simple. The calculator adds matching components. It also subtracts matching components. This helps compare positions, movements, and changes.
Products and Angles
The dot product shows how much two vectors point in the same direction. A positive dot product means the vectors mostly agree. A negative value means they mostly oppose each other. A zero value often means they are perpendicular. The angle formula uses the dot product and both magnitudes. This gives the angle in degrees and radians.
Cross Product and Area
The cross product creates a vector perpendicular to two 3D vectors. Its magnitude is also useful. It equals the area of the parallelogram formed by the two vectors. Half of that value gives the area of the related triangle. In 2D work, the z part of the cross product shows signed area behavior.
Projection and Unit Vectors
A unit vector keeps direction but changes length to one. It is useful when direction matters more than size. Projection measures how much of one vector lies along another vector. This is helpful in physics, geometry, graphics, engineering, and navigation. The calculator also finds distance between vector tips and the midpoint between them.
Practical Use
Students can use this page to check homework steps. Designers can compare directions. Developers can test graphics calculations. Engineers can review force components. The downloadable report also makes sharing easier. Always review units and signs before using final results in important work.
FAQs
What is a vector?
A vector is a quantity with magnitude and direction. It is often written with components such as x, y, and z. Common examples include force, velocity, acceleration, displacement, and position.
Can this calculator handle 2D vectors?
Yes. Select 2D mode. The calculator will use x and y components. It will treat z as zero, so dot products, magnitudes, angles, and projections still work correctly.
What does the dot product tell me?
The dot product measures directional alignment. A positive value suggests similar direction. A negative value suggests opposite direction. A zero value usually means the vectors are perpendicular.
What is the cross product used for?
The cross product creates a vector perpendicular to two 3D vectors. Its magnitude also gives parallelogram area. It is useful in geometry, physics, torque, graphics, and surface calculations.
Why is my angle undefined?
The angle is undefined when either vector has zero magnitude. A zero vector has no clear direction. Since angle depends on direction, the calculator cannot compute it safely.
What is a unit vector?
A unit vector has a length of one. It keeps the original vector direction. It is used when direction is needed without the original magnitude.
What does projection mean?
Projection shows how much of one vector lies along another vector. It is useful for resolving forces, measuring shadows, finding components, and analyzing movement along a chosen direction.
Can I export my results?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a printable report. You can also print the result page directly.