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. »
Examplesopen allclose all
Basic Examples (2)
Project the vector (5, 6, 7) onto the axis:
Project onto another vector:
Use symbolic vectors:
Assume all variables are real:
Use exact arithmetic to find the projection of u onto v:
Use machine arithmetic:
Use 20‐digit-precision arithmetic:
Projection of a complex vector onto another:
Generalizations & Extensions (1)
Use a different inner product:
Find parallel and orthogonal components of a vector:
u is the sum of the parallel and orthogonal components:
Unnormalized Gram–Schmidt algorithm (use Orthogonalize for a better implementation):
Do Gram–Schmidt on a random set of 3 vectors:
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:
Introduced in 2007
Updated in 2014