Advanced Vector Input
Enter 3D vectors A and B to measure how strongly A acts along B.
Example Data Table
This sample illustrates how the calculator interprets vector direction, magnitude, and signed projection values.
| Vector A | Vector B | Dot Product | |B| | Scalar Projection of A on B |
|---|---|---|---|---|
| (6, 2, -1) | (3, 4, 0) | 26 | 5 | 5.2 |
| (2, -3, 1) | (1, 2, 2) | -2 | 3 | -0.6667 |
| (5, 5, 5) | (1, 1, 1) | 15 | 1.7321 | 8.6603 |
Formula Used
Scalar Projection of A on B = (A · B) / |B|
A · B = axbx + ayby + azbz
|B| = √(bx2 + by2 + bz2)
Scalar projection measures the signed length of vector A in the direction of vector B. Positive values show similar direction, negative values show opposite direction, and zero suggests perpendicularity.
The calculator also computes the dot product, the magnitudes of both vectors, the unit direction of B, and the angle between vectors for deeper interpretation.
How to Use This Calculator
- Enter the x, y, and z components for vector A.
- Enter the x, y, and z components for vector B.
- Click the calculate button to process the vectors.
- Review the scalar projection, dot product, angle, and unit vector details.
- Use CSV or PDF export to save the current result.
Why Scalar Projection Matters
Scalar projection is useful in vector geometry, mechanics, graphics, and machine learning. It helps isolate how much one quantity contributes in a chosen direction, which is especially valuable when studying components of force, displacement, velocity, or feature direction.
Because the result is signed, it also reveals directional agreement. A large positive value means strong alignment, while a large negative value indicates strong opposition along the reference vector.
FAQs
1. What does scalar projection measure?
It measures the signed length of one vector along another vector’s direction. It shows how much of vector A acts parallel to vector B.
2. Can scalar projection be negative?
Yes. A negative result means vector A points partly opposite the direction of vector B, even though both still occupy the same vector space.
3. Why is the magnitude of B in the denominator?
Dividing by |B| normalizes the reference direction. That converts the dot product into a signed component length instead of a scale-dependent product.
4. What happens if vector B is zero?
The scalar projection becomes undefined because a zero vector has no direction. The calculator blocks that case and shows an error.
5. Is this different from vector projection?
Yes. Scalar projection is only the signed magnitude. Vector projection includes both magnitude and direction, producing a full vector result.
6. Does the calculator work for perpendicular vectors?
Yes. Perpendicular vectors have a dot product of zero, so the scalar projection is also zero, indicating no component along B.
7. Why does the tool show the angle too?
The angle helps interpret alignment. Acute angles give positive projections, obtuse angles give negative projections, and right angles give zero projection.
8. Where is scalar projection used in practice?
It appears in physics, engineering, robotics, graphics, optimization, and data science whenever directional contribution or signed component length matters.