How do you find the coordinate of a vector with the unit vector?

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SUMMARY

The discussion focuses on determining the coordinates of a vector using a unit vector. It establishes that if one coordinate, specifically ##x_1##, is known, the other coordinate ##x_2## can be derived from the equation ##x_2e_2=x-x_1e_1##. Additionally, it highlights the use of inner products to find the coordinate ##\langle e_2,x\rangle##, providing an alternative method for calculation.

PREREQUISITES
  • Understanding of vector representation in the form ##x=x_1e_1+x_2e_2##
  • Familiarity with unit vectors and their properties
  • Knowledge of inner products in vector spaces
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study vector decomposition techniques in linear algebra
  • Learn about inner products and their applications in vector calculations
  • Explore the properties of unit vectors in multi-dimensional spaces
  • Investigate coordinate transformations and their implications in vector analysis
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who require a deeper understanding of vector analysis and coordinate systems.

BlueRope
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If you already have one of the two coordinates?
 
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If ##x=x_1e_1+x_2e_2## and you know the value of ##x_1##, you can get the value of ##x_2## from ##x_2e_2=x-x_1e_1##.

Another option (if you understand inner products) is to just compute ##\langle e_2,x\rangle##.
 

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