Parallel/Orthogonal in Calc 3.

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SUMMARY

The discussion focuses on decomposing vector 'w'=<40,-19,-1> into two components: 'g', which is parallel to vector 'v'=<1,-1,5>, and 'h', which is orthogonal to 'v'. The correct approach involves using the dot product to find 'g' as a scalar multiple of 'v' and calculating 'h' as the difference between 'w' and 'g'. The solution requires understanding the properties of dot products and vector projections.

PREREQUISITES
  • Understanding of vector decomposition
  • Familiarity with dot products
  • Knowledge of vector projections
  • Basic skills in linear algebra
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  • Learn how to calculate vector projections using the formula for projection of 'w' onto 'v'
  • Study the properties of dot products and their geometric interpretations
  • Explore orthogonal vectors and their significance in vector spaces
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Students studying linear algebra, particularly those focusing on vector decomposition and applications of dot products in higher-dimensional spaces.

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Homework Statement


Let vector 'v'=<1,-1,5> and vector 'w'=<40,-19,-1>. Represent 'w' in the form of 'w'='g'+'h',
where 'g' is parallel to 'v', and 'h' is orthogonal to 'v'. This is related to the dot product. But I don't know where to begin




Homework Equations


'v' is a vector. 'g' is a vector. 'w' is a vector. 'h' is a vector


The Attempt at a Solution


So far I set up an equation <40,-19,-1>='g'+'h'
I doubt this is right so far. I know that g has to be a multiple of v and h has to = 0 in order for it to be orthogonal.
 
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well what do you know about dot products? in particular consider
w \bullet v
 

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