Project one vector onto another, showing the projection vector, scalar projection, and step-by-step work.
Dimension
Vector 1 (to project)
Vector 2 (onto)
What is vector projection?
The vector projection of v₁ onto v₂ gives the component of v₁ that lies along v₂. It answers: how much of v₁ points in the direction of v₂?
Scalar vs vector projection
The scalar projection (component) is a signed length, while the vector projection is a vector in the direction of v₂ with that length.
Result
The projection of [3, 4] onto [1, 0] is [3, 0].
Steps
v₁ = [3, 4], v₂ = [1, 0]
v₁ · v₂ = (3)(1) + (4)(0) = 3
v₂ · v₂ = x²=1 + y²=0 = 1
|v₂| = √1 = 1
Scalar projection = (v₁ · v₂) / |v₂| = 3 / 1 = 3
Scale factor = (v₁ · v₂) / (v₂ · v₂) = 3 / 1 = 3
proj = 3 × [1, 0] = [3, 0]
Also in Linear Algebra
Linear Algebra
A vector projection calculator finds the component of one vector that lies along the direction of another. It computes both the vector projection and the scalar projection.
The vector projection of v₁ onto v₂ is proj = ((v₁·v₂)/(v₂·v₂)) × v₂. The scalar projection (component) is comp = (v₁·v₂)/|v₂|.
The scalar projection tells you how much of v₁ lies along v₂. If negative, v₁ points partly opposite to v₂.
proj_v₂(v₁) = ((v₁·v₂)/(v₂·v₂)) × v₂
Vector projection formula.
Frequently asked questions
The projection is the zero vector because there is no component of v₁ in the direction of v₂.
No — projection onto the zero vector is undefined because the zero vector has no direction.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Calculate the dot product of two vectors in 2D or 3D, with angle and orthogonality detection.
Perform vector operations including addition, subtraction, dot product, cross product, and scalar multiplication.
Calculate the angle between two vectors in 2D or 3D using the dot product formula.
Normalize a 2D or 3D vector to find its unit vector (magnitude 1) with step-by-step work.
