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Generated on: July 20, 2026 at 09:33 PM

Vector Projection Calculator

Inputs

a — x-component3
a — y-component4
a — z-component0
b — x-component1
b — y-component0
b — z-component0

Generated with OneCalc · https://onecalc.app/en/math/vector-projection

Mathematics

Vector Projection Calculator

Project vector a onto vector b: scalar projection, vector projection components, and the perpendicular rejection. Works in 2D (leave z = 0) and 3D, with step-by-step derivations.

Inputs

Vector a (projected)

Vector b (direction)

Results

Enter a value to see results.

Details

Step-by-step derivation

\begin{aligned} \operatorname{comp}_{\mathbf{b}}\mathbf{a} &= \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \\ &= \frac{?}{?} \\ &= ? \end{aligned}
\begin{aligned} p_x &= \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}}\, b_x \\ &= \frac{?}{{\text{?}}}(1) \\ &= ? \end{aligned}
\begin{aligned} p_y &= \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}}\, b_y \\ &= \frac{?}{{\text{?}}}(0) \\ &= ? \end{aligned}
\begin{aligned} p_z &= \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}}\, b_z \\ &= \frac{?}{{\text{?}}}(0) \\ &= ? \end{aligned}
\begin{aligned} r_x &= a_x - p_x \\ &= 3 - (?) \\ &= ? \end{aligned}
\begin{aligned} r_y &= a_y - p_y \\ &= 4 - (?) \\ &= ? \end{aligned}
\begin{aligned} r_z &= a_z - p_z \\ &= 0 - (?) \\ &= ? \end{aligned}

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Vector projection

Projecting a vector a onto a vector b answers a simple question: how much of a points in the direction of b? The answer comes in two flavours — a single number (the scalar projection) and a full vector (the vector projection). Together with the rejection, they split a into a part along b and a part perpendicular to it.

Scalar projection

The scalar projection of a onto b is the signed length of a measured along the direction of b:

comp⁡ba=a⋅b∣b∣\operatorname{comp}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}

It is positive when a leans the same way as b, zero when they are perpendicular, and negative when a leans the opposite way.

Vector projection

The vector projection is a vector that points along b and has length equal to the absolute value of the scalar projection:

proj⁡ba=a⋅bb⋅b b=a⋅b∣b∣2 b\operatorname{proj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}}\,\mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2}\,\mathbf{b}

The coefficient a⋅bb⋅b\frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} rescales b so that the result lands directly "under" the tip of a.

The rejection

The rejection is whatever is left of a after the projection is removed:

rej⁡ba=a−proj⁡ba\operatorname{rej}_{\mathbf{b}}\mathbf{a} = \mathbf{a} - \operatorname{proj}_{\mathbf{b}}\mathbf{a}

It is perpendicular to b, so the projection and the rejection are orthogonal and add back to the original vector: a=proj⁡ba+rej⁡ba\mathbf{a} = \operatorname{proj}_{\mathbf{b}}\mathbf{a} + \operatorname{rej}_{\mathbf{b}}\mathbf{a}.

Worked example

Let a = (3, 4, 0) and b = (1, 0, 0).

First the dot products:

a⋅b=(3)(1)+(4)(0)+(0)(0)=3,b⋅b=12+02+02=1\mathbf{a} \cdot \mathbf{b} = (3)(1) + (4)(0) + (0)(0) = 3, \qquad \mathbf{b} \cdot \mathbf{b} = 1^2 + 0^2 + 0^2 = 1

The magnitude of b is ∣b∣=1=1|\mathbf{b}| = \sqrt{1} = 1, so the scalar projection is:

comp⁡ba=31=3\operatorname{comp}_{\mathbf{b}}\mathbf{a} = \frac{3}{1} = 3

The vector projection uses the coefficient 31=3\frac{3}{1} = 3:

proj⁡ba=3 (1,0,0)=(3,0,0)\operatorname{proj}_{\mathbf{b}}\mathbf{a} = 3\,(1, 0, 0) = (3, 0, 0)

And the rejection is the perpendicular remainder:

rej⁡ba=(3,4,0)−(3,0,0)=(0,4,0)\operatorname{rej}_{\mathbf{b}}\mathbf{a} = (3, 4, 0) - (3, 0, 0) = (0, 4, 0)

As expected, (3, 0, 0) and (0, 4, 0) are perpendicular and sum back to (3, 4, 0).

2D and 3D

The same formulas work in any dimension. For a flat 2D problem, leave the z-components at 0 and read off the x and y results; the z output stays 0.

The work analogy

In physics, the work done by a constant force is the dot product of force and displacement:

W=F⋅d=∣F∣ ∣d∣cos⁡θW = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}|\,|\mathbf{d}|\cos\theta

This is the force magnitude times the scalar projection of the displacement onto the force. Only the part of the force aligned with the motion does work — the perpendicular part (the rejection) contributes nothing. The same idea resolves the weight of an object on a ramp into a component sliding it down the slope and a component pressing it into the surface.

Applications

  • Gram–Schmidt orthogonalization: subtract the projection of each vector onto the previous ones to build an orthogonal basis.
  • Least-squares fitting: the best approximation of a point in a subspace is its projection onto that subspace.
  • Computer graphics: projecting a light or view direction onto a surface drives shading and shadow length.
  • Resolving forces: splitting a force into components along and across a direction, as on an inclined plane.

Frequently Asked Questions (FAQ)

What is the difference between scalar and vector projection?

The scalar projection comp_b(a) = (a · b) / |b| is a single signed number: how far a reaches along the direction of b. The vector projection proj_b(a) = ((a · b) / (b · b)) b is a full vector — it points along b and has length equal to the absolute value of the scalar projection. In short, the scalar projection answers "how much," and the vector projection answers "how much, and in which direction."

How do you project one vector onto another?

Compute the dot product a · b and the squared magnitude b · b = |b|². The scalar projection is (a · b) / |b|. The vector projection is the coefficient (a · b) / (b · b) multiplied by b. For a = (3, 4) and b = (1, 0): a · b = 3, |b| = 1, so the scalar projection is 3 and the vector projection is (3, 0).

What is the vector rejection?

The rejection is the part of a left over after removing its projection onto b: a − proj_b(a). It is perpendicular to b, so the projection and the rejection split a into two orthogonal pieces that add back to a. This decomposition is the basis of the Gram–Schmidt process and of resolving a force into components along and across a surface.

Where are projections used in physics?

Work done by a constant force is W = F · d = |F| |d| cos θ, which is the force magnitude times the scalar projection of the displacement along the force (or vice versa). Only the component of the force along the direction of motion does work; the perpendicular component does none. Projections also resolve weight into components along and normal to an inclined plane.

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