Find two unit vectors orthogonal to both given vectors

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SUMMARY

The discussion focuses on finding two unit vectors orthogonal to the vectors i + j + k and 3i + k using the cross product method. The user calculated the cross product, resulting in the vector i + 2j - 3k, and determined its magnitude to be √14. The user rationalized the denominator to arrive at two unit vectors: <-√(14)/14 i, -√(14)/7 j, 3√(14)/14 k> and <√(14)/14 i, √(14)/7 j, -3√(14)/14 k>. Verification of the results involves checking that the dot products with the original vectors equal zero and confirming that both vectors have a length of 1.

PREREQUISITES
  • Understanding of vector operations, specifically cross products
  • Knowledge of unit vectors and their properties
  • Familiarity with calculating magnitudes of vectors
  • Ability to perform dot product calculations
NEXT STEPS
  • Study the properties of cross products in three-dimensional space
  • Learn how to verify orthogonality using dot products
  • Explore the concept of unit vectors and their significance in vector analysis
  • Practice problems involving vector normalization and rationalization of denominators
USEFUL FOR

Students studying linear algebra, particularly those focusing on vector calculus and geometry. This discussion is beneficial for anyone needing to understand vector orthogonality and unit vector calculations.

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Hello,

Could someone please review my work and see if it is correct. Thanks :smile:

Homework Statement



Find two unit vectors orthogonal to both given vectors.
i + j + k, 3i + k

Homework Equations





The Attempt at a Solution



So I used cross product and got A x B= i+2j-3k

Then I got magnitude: to be √ (14)

problem asks to rationalize denominator and I eventually get..

lower values: <-√(14)/14 i, -√(14)/7 j, 3√(14)/14 k>

larger values: <√(14)/14 i, √(14)/7 j, -3√(14)/14 k>



I just want to make sure that I did everything correctly.
 
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Looks right to me. You can verify your answers by checking that their inner (dot) products with both given vectors are zero, and that their lengths are 1.

You really only need to check one answer because you know the second answer has to be the negative of the first.
 

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