Vector Scaling Inputs
Formula Used
Basic scaling: k\vec{v} = (k v_1, k v_2, \dots, k v_n)
Magnitude: |\vec{v}| = √(v₁² + v₂² + ... + vₙ²)
Target magnitude scalar: k = target magnitude ÷ original magnitude
Direction rule: positive scalars preserve direction, negative scalars reverse direction, and zero produces the zero vector.
This calculator applies the scalar to each vector component individually. It then recomputes magnitude, unit vector values, dot product, and the angle between the original and scaled vectors.
In percentage mode, a factor of 150% means multiplying every component by 1.5. In target magnitude mode, the calculator finds the exact scalar required to reach the requested length.
How to Use This Calculator
- Select the vector dimension from 2D to 6D.
- Enter the vector components in the same order as the chosen dimension.
- Choose direct scalar, percentage factor, or target magnitude mode.
- Fill only the input needed for the selected mode.
- Set your preferred precision and optional display unit.
- Press Scale Vector to place the result above the form.
- Review the component table, magnitude changes, and direction status.
- Use the CSV or PDF buttons to save the output.
Example Data Table
| Case | Input vector | Mode | Factor | Scaled vector | Magnitude change |
|---|---|---|---|---|---|
| Geometry | (3, 4) | Direct scalar | 2 | (6, 8) | 5 to 10 |
| Physics | (2, -1, 5) | Percentage | 75% | (1.5, -0.75, 3.75) | Reduced by 25% |
| Target length | (1, 2, 2) | Target magnitude | 6 | (2, 4, 4) | 3 to 6 |
| Reversal | (4, -3) | Direct scalar | -1.5 | (-6, 4.5) | Direction reversed |
Frequently Asked Questions
1. What does vector scaling mean?
Vector scaling multiplies every component by the same scalar. The vector keeps its component ratios, while its length changes according to the scalar’s absolute value.
2. Does scaling change vector direction?
A positive scalar keeps the same direction. A negative scalar flips the vector to the opposite direction. A zero scalar collapses the vector to the origin.
3. Why does target magnitude need a nonzero vector?
A zero vector has no direction, so it cannot be stretched proportionally to a chosen magnitude. The calculator blocks that case to avoid undefined scaling.
4. What is the difference between direct and percentage scaling?
Direct scaling uses the scalar exactly as entered. Percentage scaling converts the entered percentage into a multiplier, so 150% becomes 1.5 and 25% becomes 0.25.
5. Why is the angle sometimes 0° or 180°?
Scaling preserves linear alignment. Positive factors keep vectors parallel, producing 0°. Negative factors reverse them, producing 180°, unless the result becomes the zero vector.
6. Can I use decimals and negative values?
Yes. The calculator accepts decimals, fractions written as decimals, and negative components. This makes it useful for coordinate geometry, forces, velocity, and data modelling.
7. What unit should I enter?
The unit field is only a display label for your result panel. Use metres, newtons, pixels, or any consistent unit that matches the original vector context.
8. What does the dot product show here?
The dot product helps confirm alignment between the original and scaled vectors. Its sign also reflects whether the scaling kept or reversed the original direction.