Free · Vectors
Orthogonal Vector Projection Components
Project one vector onto another in two or three dimensions, including signed components and captured inverse relationships.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
For a=(3,4) and b=(2,0), the factor is1.5 and the projection is(3,0).
proj_b(a) = ((a·b)/(b·b)) b; the residual a−proj_b(a) is perpendicular to b.
Worked example
Enter these known values and leave the other values blank.
- Number of dimensions
- 2D
- x
- 3
- y
- 4
- x
- 2
- y
- 0
- x
- 3
- y
- 0
- Projection factor
- 1.5
Assumptions and limitations
- Choose2D or3D and enter components relative to the same orthonormal axes. Components are signed real numbers; the target vector b must be nonzero.
- The projection factor is a·b divided by b·b. It is not the scalar projection a·b/|b|. A zero or negative factor is valid.
- All captured inverse variants remain available. A quadratic inverse for an unknown b component selects the captured positive-square-root algebraic branch; there can be another valid component. Zero divisors may make an inverse underdetermined or impossible. Clear an unknown rather than overdetermining the inputs.
- The diagram is a fixed geometric illustration of the projection and perpendicular residual; it does not plot your entered vectors.
Common questions
Can the projected vector be zero?
Yes. It is zero if a is zero or a is perpendicular to the nonzero target vector b.
What does a negative factor mean?
The projection points opposite to b. A negative component or factor is not an invalid vector.
References
Orthogonal projection onto a vector divides by its squared norm; the target vector must be nonzero. Signed components and a zero projected vector are allowed. Preserve both captured dimensions and all inverse variants.