Find vectors that produce certain orthogonal projection

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SUMMARY

The discussion centers on calculating the original vector w from its orthogonal projection v onto a plane spanned by vectors u1 and u2 in three-dimensional space. The formula provided for this calculation is w = \frac{}{}u_1 + \frac{}{}u_2. This formula utilizes the inner product to derive the components of w based on the known vectors v, u1, and u2.

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  • Understanding of vector spaces and orthogonal projections
  • Familiarity with inner product notation
  • Basic knowledge of three-dimensional geometry
  • Proficiency in LaTeX for mathematical expressions
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  • Study the properties of orthogonal projections in vector spaces
  • Learn about inner product spaces and their applications
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Mathematicians, physics students, and anyone involved in computational geometry or linear algebra who needs to understand vector projections and their calculations.

fmilano
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I have vector [ tex ] v [ /tex ] produced by an orthogonal projection of vector [ tex ] w [ /tex ] over plane spanned by vectors [ tex ] u_1 [ /tex ] and [ tex ] u_2 [ /tex ], in a three dimensional space. If I know [ tex ] v [ /tex ], [ tex ] u_1 [ /tex ] and [ tex ] u_2 [ /tex ], how could I calculate [ tex ] w [ /tex ]?

Thanks in advance,

Federico
 
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So you want to calculate the orthogonal projection on the space spanned by u1 and u2.

There is a formula for that:

[tex]w=\frac{<v,u_1>}{<u_1,u_1>}u_1+\frac{<v,u_2>}{<u_2,u_2>}u_2[/tex]

To write in LaTeX, you don't need to write the space in [ tex ] and [ /itex ]
 

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