# Projection

Projection[u,v]

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

Projection[u,v,f]

finds projections with respect to the inner product function f.

# Details

• 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. »
• Projection[u,v,Dot] effectively assumes that all elements of u and v are real. »

# Examples

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## Basic Examples(2)

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

Project onto another vector:

## Scope(9)

Find the projection of a machine-precision vector onto another:

Projection of a complex vector onto another:

Projection of an exact vector onto another:

Projection of an arbitrary-precision vector onto another:

The projection of large numerical vectors is computed efficiently:

Project symbolic vectors:

Give an inner product of Dot to assume all expressions are real-valued:

Project vectors that are not lists using an explicit inner product:

Specify the inner product using a pure function:

## 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 :

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

Wolfram Research (2007), Projection, Wolfram Language function, https://reference.wolfram.com/language/ref/Projection.html (updated 2014).

#### Text

Wolfram Research (2007), Projection, Wolfram Language function, https://reference.wolfram.com/language/ref/Projection.html (updated 2014).

#### BibTeX

@misc{reference.wolfram_2021_projection, author="Wolfram Research", title="{Projection}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Projection.html}", note=[Accessed: 31-July-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2021_projection, organization={Wolfram Research}, title={Projection}, year={2014}, url={https://reference.wolfram.com/language/ref/Projection.html}, note=[Accessed: 31-July-2021 ]}

#### CMS

Wolfram Language. 2007. "Projection." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/Projection.html.

#### APA

Wolfram Language. (2007). Projection. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Projection.html