Vector Projection Calculator
Compute vector projection easily: enter two vectors, see dot product, magnitude squared, scalar coefficient, and projected vector with steps instantly.
Equation Preview
Helping Notes
- Use same length for a and b. Mixed dimensions are invalid.
- Scalar coefficient k = (a·b) / ‖b‖²; projected vector = k · b.
- b must be non-zero; otherwise ‖b‖² = 0 and projection is undefined.
- Separators: commas, spaces, or new lines. Decimals and negatives allowed.
Results
Projected Vector projb(a)
Scalar & Dot/Norm Details
Decomposition
Error
What is Vector Projection Calculator?
A Vector Projection Calculator determines the component of a vector \(\mathbf a\) that lies along another vector \(\mathbf b\). The result is a vector pointing parallel to \(\mathbf b\) with the appropriate length; a related result is the scalar projection, the signed length of \(\mathbf a\) along \(\mathbf b\). These quantities are fundamental in geometry, physics (work, forces), computer graphics, optimization, and data analysis. The core formulas are:
About the Vector Projection Calculator
This tool accepts 2D, 3D, or n‑dimensional vectors and outputs the vector projection, scalar projection, perpendicular component, and the angle between vectors. It displays clean, step‑by‑step working: dot product, magnitudes, ratio \((\mathbf a\cdot\mathbf b)/(\lVert\mathbf b\rVert^2)\), and the final projection vector. Inputs may be integers, decimals, or fractions; order matters because projecting \(\mathbf a\) onto \(\mathbf b\) differs from projecting \(\mathbf b\) onto \(\mathbf a\). The interface also reports edge cases—for example, when \(\mathbf b=\mathbf 0\), the projection is undefined—and suggests corrections such as choosing a nonzero reference vector. All formulas render responsively so they remain readable on phones, tablets, and desktops.
How to Use this Vector Projection Calculator
- Enter the components of \(\mathbf a\) and \(\mathbf b\) (comma‑separated for higher dimensions).
- Choose whether you also want the perpendicular component and angle.
- Click calculate to compute dot product, magnitudes, and the projection results.
- Review the formatted steps and confirm the direction and length make sense for your context.
- Adjust inputs to compare \(\operatorname{proj}_{\mathbf b}\mathbf a\) with \(\operatorname{proj}_{\mathbf a}\mathbf b\) if desired.
Examples
Example 1: 2D projection
Let \(\mathbf a=(3,4)\), \(\mathbf b=(5,0)\).
Example 2: 3D projection
Let \(\mathbf a=(1,2,3)\), \(\mathbf b=(-2,1,2)\).
FAQs
What is a vector projection in plain terms?
It’s the part of one vector that lies along another vector’s direction, represented as a new vector.
What’s the difference between vector and scalar projection?
Vector projection is a vector parallel to the reference; scalar projection is the signed length along that direction.
Can this handle n‑dimensional inputs?
Yes. Enter as many components as needed; operations generalize to any finite dimension.
What happens if the reference vector is zero?
The projection is undefined because division by its squared magnitude would be required.
How do I find the perpendicular component?
Subtract the projection from the original vector: \(\mathbf a_{\perp}=\mathbf a-\operatorname{proj}_{\mathbf b}\mathbf a\).
Does the order matter?
Yes. \(\operatorname{proj}_{\mathbf b}\mathbf a\) generally differs from \(\operatorname{proj}_{\mathbf a}\mathbf b\).
How is the angle related to projection?
Through the dot product identity \(\cos\theta=(\mathbf a\cdot\mathbf b)/(\lVert\mathbf a\rVert\lVert\mathbf b\rVert)\); projection length equals \(\lVert\mathbf a\rVert\cos\theta\).
Do units carry through?
Yes. The projection keeps the same physical units as the original vector.
Can I enter fractions or decimals?
Both work. The calculator computes exact rational forms where possible or decimal approximations otherwise.
Is the projection length always ≤ the original length?
Yes. By definition, \(|\operatorname{comp}_{\mathbf b}\mathbf a|\le\lVert\mathbf a\rVert\).
Where is this used in practice?
Decomposing forces, finding work, separating signals, shading in graphics, and orthogonal decompositions in linear algebra.
What about numerical precision?
For very large or tiny values, rounding can affect results slightly; reviewing exact forms may help.