Math July 13, 2026 · 8 Min Read

Vector Projection Calculator – Guide & Formulas

Calculate the scalar and vector projection of one vector onto another. Get the projection component and rejection vector with step-by-step solutions.

Calculate the scalar projection (component) and vector projection of one vector onto another. Also find the rejection vector (perpendicular component).

Key Takeaway

Use the free Vector Projection Calculator to calculate the scalar and vector projection of one vector onto another. get the projection component and rejection vector with step-by-step solutions. Get instant results with step-by-step explanations.

How to Use the Vector Projection Calculator

  1. Enter the components of the vector to project (a⃗).
  2. Enter the components of the vector to project onto (b⃗).
  3. Click "Calculate" to see the scalar projection, vector projection, and rejection.
  4. Review the step-by-step dot product and magnitude calculations.

The Formula

proj_b(a) = ((a·b) / |b|²) × b | comp_b(a) = (a·b) / |b|

Variable Definitions

  • proj_b(a): Vector projection of a onto b (a vector parallel to b)
  • comp_b(a): Scalar projection of a onto b (the component of a along b)
  • a⃗ · b⃗: Dot product of vectors a and b
  • |b⃗|: Magnitude of vector b
  • rejection: The perpendicular component: a⃗ - proj_b(a)

Example: Project A = (3, 4) onto B = (1, 0)

Find the vector and scalar projection of A = (3, 4) onto B = (1, 0).

  1. Step 1: Calculate dot product: A·B = (3)(1) + (4)(0) = 3.
  2. Step 2: Calculate |B|² = 1² + 0² = 1.
  3. Step 3: Scalar projection = A·B / |B| = 3 / 1 = 3.
  4. Step 4: Vector projection = (3/1) × (1, 0) = (3, 0).
  5. Step 5: Rejection = A - proj = (3,4) - (3,0) = (0, 4).

Frequently Asked Questions

What is vector projection?

Vector projection of A onto B is the component of A that lies along the direction of B. It produces a vector parallel to B.

What is the difference between scalar and vector projection?

Scalar projection gives a single number (the length of the projection). Vector projection gives a vector with both magnitude and direction along B.

What is the rejection vector?

The rejection (or perpendicular component) is the part of A that is perpendicular to B. It is calculated as A - proj_B(A).

How is projection related to the dot product?

The scalar projection uses the dot product: comp_B(A) = (A·B)/|B|. The dot product measures how much A points in the direction of B.

What happens if B is a unit vector?

If B is a unit vector (|B| = 1), the projection formula simplifies to proj_B(A) = (A·B) × B, since |B|² = 1.

Can projection be negative?

Yes, the scalar projection can be negative if the angle between A and B is greater than 90°. This means A points partly opposite to B.

What is orthogonal projection?

Orthogonal projection is another name for vector projection. It projects a vector perpendicularly onto another vector or subspace.

How do I project onto a plane?

To project a vector onto a plane, first find the normal vector to the plane. Then subtract the projection onto the normal from the original vector.

What are real-world uses of vector projection?

Vector projection is used in physics (finding force components), computer graphics (shadows and lighting), and engineering (decomposing forces).

What is the relationship between projection and rejection?

The original vector equals the sum of its projection and rejection: A = proj_B(A) + rejection_B(A). They are perpendicular to each other.