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Orthogonal Projection Calculator

Calculates the projection of vector v onto the subspace spanned by vector u.

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Scalar Coefficient

Formula first

Overview

The orthogonal projection of a vector v onto a vector u determines the component of v that points in the same direction as u. This process effectively maps v onto the line spanned by u, creating a new vector that is the closest point in that line to the original vector v.

Symbols

Variables

c = Scalar Coefficient, u v = u · v, u u = u · u

Scalar Coefficient
Variable
u · v
Variable
u · u
Variable

Apply it well

When To Use

When to use: Use this formula when you need to decompose a vector into parallel and perpendicular components relative to a reference vector. It is essential in the Gram-Schmidt process for building orthonormal bases and for finding the shortest distance from a point to a line.

Why it matters: Orthogonal projections are the mathematical foundation for linear regression in statistics, signal processing, and computer graphics. They allow engineers to resolve forces into specific directions and data scientists to reduce the dimensionality of complex datasets.

Avoid these traps

Common Mistakes

  • Using the magnitude of u instead of the dot product u · u (the squared magnitude) in the denominator.
  • Confusing the vector being projected (v) with the vector defining the direction (u).

One free problem

Practice Problem

In a physics simulation, a force vector v is projected onto a directional vector u. If the dot product u ⋅ v is calculated as 18 and the dot product of u with itself (u ⋅ u) is 6, what is the resulting scalar multiplier for the projection?

u · v18
u · u6

Solve for: result

Hint: Divide the dot product of the two vectors by the dot product of the reference vector u with itself.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Linear Algebra and Its Applications by David C. Lay
  2. Introduction to Linear Algebra by Gilbert Strang
  3. Wikipedia: Vector projection
  4. Wikipedia: Projection (linear algebra)
  5. Lay, David C. Linear Algebra and Its Applications. 5th ed. Pearson, 2016.
  6. Wikipedia: Projection (linear algebra). Wikimedia Foundation. Available at: https://en.wikipedia.org/wiki/Projection_(linear_algebra)
  7. Lay, D. C., Lay, S. R., & McDonald, J. J. (2016). Linear Algebra and Its Applications (5th ed.). Pearson.