Vector Projection Length Calculator

Measure vector alignment quickly with clear two-vector inputs. Review signed results, angles, and projected components. Build stronger intuition for geometry, motion, force, and modeling.

Calculator Inputs

Use two vectors in 2D or 3D to evaluate alignment and projection length.

Example Data Table

Case Vector A Vector B Dot Product Projection Length Interpretation
Example 1 (6, 2, 3) (4, 1, 2) 32 6.983 Strong positive alignment.
Example 2 (5, -1) (2, 2) 8 2.828 Moderate forward projection.
Example 3 (3, 4, 0) (0, 5, 0) 20 4.000 Projection matches y-direction component.

Formula Used

Vector projection length measures how much of vector A lies along vector B. The signed scalar projection keeps direction. The absolute projection length reports the size only.

dot(A, B) = A_xB_x + A_yB_y + A_zB_z |B| = sqrt(B_x^2 + B_y^2 + B_z^2) signed projection of A onto B = dot(A, B) / |B| projection length = |dot(A, B) / |B|| proj_B(A) = (dot(A, B) / |B|^2) x B

If the signed result is positive, A points generally with B. If negative, A points partly against B. A zero result means they are perpendicular.

How to Use This Calculator

  1. Choose whether your vectors use 2D or 3D coordinates.
  2. Enter all components for vector A and vector B.
  3. Add a unit label, such as m, cm, or units.
  4. Set the number of decimals you want in the output.
  5. Press Calculate Projection to show the result above the form.
  6. Review projection length, signed projection, angle, and projection vector.
  7. Use the CSV or PDF buttons to save your results.

Frequently Asked Questions

1. What is vector projection length?

It is the amount of one vector that lies along another vector's direction. It can be reported as a signed scalar projection or as an absolute length.

2. Why does the calculator need vector B to be nonzero?

Projection onto a zero vector is undefined because the direction does not exist. The formula divides by the magnitude of vector B, so that magnitude cannot be zero.

3. What does a negative signed projection mean?

A negative signed projection means vector A points partly opposite to vector B. The absolute projection length still shows the size of the aligned component.

4. Can I use this for 2D and 3D vectors?

Yes. Select 2D to hide z-components or choose 3D to include them. The same projection ideas apply in either coordinate system.

5. What is the difference between projection length and projection vector?

Projection length is a scalar size. Projection vector is the full vector that lies along B and includes direction and coordinates for the aligned component.

6. Where is vector projection used?

It appears in geometry, physics, graphics, machine learning, signal analysis, and engineering. Common uses include resolving forces, shadows, distances, and directional similarity.

7. Why does the angle matter here?

The angle controls how much of A points along B. Smaller angles give larger positive projections, ninety degrees gives zero, and obtuse angles give negative signed projections.

8. Does the unit label affect the math?

No. The unit label only helps present results clearly. The numerical calculation depends only on the vector components you enter.