FREE TOOLS Vector Projection Calculator 

What Is a FREE Vector Projection Calculator?

A FREE Vector Projection Calculator is an online math tool that calculates the projection of one vector onto another vector. It is commonly used in linear algebra, geometry, physics, engineering, and computer science.

The vector projection formula is:

proj⁡b(a)=a⋅b∥b∥2b\operatorname{proj}_{\mathbf b}(\mathbf a)=\frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf b\rVert^2}\mathbf b

a=p+r,r⊥b\mathbf a=\mathbf p+\mathbf r,\qquad \mathbf r\perp\mathbf ba=p+r,r⊥b

comp⁡b(a)=6cos⁡(55∘)=3.4\operatorname{comp}_{\mathbf b}(\mathbf a)=6\cos(55^\circ)=3.4compb​(a)=6cos(55∘)=3.4

The acute angle gives a positive component, so the projection points with b

θ\thetaθ

°

θ\thetaθθ = 55°abpr

Give feedback proj⁡b(a)=a⋅b∥b∥2b\operatorname{proj}_{\mathbf b}(\mathbf a) = \frac{\mathbf a\cdot\mathbf b}{\|\mathbf b\|^2}\mathbf b

Where:

• a\mathbf a = vector being projected
• b\mathbf b = vector onto which a\mathbf a is projected
• a⋅b\mathbf a\cdot\mathbf b = dot product
• ∥b∥\|\mathbf b\| = magnitude of vector b\mathbf b

Example

Suppose: a=(3,4)\mathbf a=(3,4)

and b=(1,0)\mathbf b=(1,0)

The projection of a\mathbf a onto b\mathbf b is: proj⁡b(a)=(3,0)\operatorname{proj}_{\mathbf b}(\mathbf a)=(3,0)

So the component of vector a in the direction of vector b is: (3,0)\boxed{(3,0)}

What Can a Free Vector Projection Calculator Do?

A Vector Projection Calculator can help you:

• Calculate the projection of one vector onto another
• Find scalar and vector projections
• Work with 2D and 3D vectors
• Calculate dot products and vector magnitudes
• Determine the component of a vector in a specific direction
• Solve linear algebra and geometry problems
• Analyze forces and other vector quantities in physics

Why Is Vector Projection Useful?

Vector projection is useful when a vector needs to be separated into components parallel and perpendicular to another vector. For example, physics uses vector projections to analyze forces acting along particular directions.

In simple terms: a FREE Vector Projection Calculator determines how much of one vector points in the direction of another vector and calculates the corresponding projected vector.

Keep working with vector projections