finds the projection of the vector u onto the vector v.


finds projections with respect to the inner product function f.


  • For ordinary real vectors u and v, the projection is taken to be .
  • For ordinary complex vectors u and v, the projection is taken to be , where is Conjugate[v].
  • In Projection[u,v,f], u and v can be any expressions or lists of expressions for which the inner product function f applied to pairs yields real results. »


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Basic Examples  (2)

Project the vector (5, 6, 7) onto the axis:

Project onto another vector:

Scope  (4)

Use symbolic vectors:

Assume all variables are real:

Use exact arithmetic to find the projection of u onto v:

Use machine arithmetic:

Use 20digit-precision arithmetic:

Projection of a complex vector onto another:

Generalizations & Extensions  (1)

Use a different inner product:

Applications  (2)

Find parallel and orthogonal components of a vector:

u is the sum of the parallel and orthogonal components:

Unnormalized GramSchmidt algorithm (use Orthogonalize for a better implementation):

Do GramSchmidt on a random set of 3 vectors:

Verify orthogonality:

Generate some orthogonal polynomials:

Properties & Relations  (3)

The projection of u onto v is in the direction of v:

For ordinary vectors u and v, the projection is taken to be ( TemplateBox[{v}, Conjugate].u)/(TemplateBox[{v}, Conjugate].v)v:

The projection of u onto v is equivalent to multiplication by an outer product matrix:

Introduced in 2007
Updated in 2014