SUMMARY
The Orthonormal Basis of the plane defined by the equation x - 4y - z = 0 is derived from two independent vectors. The process begins by identifying vectors that satisfy the equation, specifically (4, 1, 0) and (1, 0, 1). These vectors are then orthogonalized using the Gram-Schmidt process and normalized to yield the final orthonormal basis vectors: e1 = (1/sqrt(2), 0, 1/sqrt(2)) and e2 = (2/3, 1/3, -2/3). The dot product of these vectors confirms their orthogonality.
PREREQUISITES
- Understanding of vector equations and their geometric interpretations.
- Familiarity with the Gram-Schmidt orthogonalization process.
- Knowledge of vector normalization techniques.
- Basic linear algebra concepts, including dot products and spans.
NEXT STEPS
- Study the Gram-Schmidt process in detail to understand vector orthogonalization.
- Learn about vector normalization and its applications in linear algebra.
- Explore the geometric interpretation of planes and their bases in three-dimensional space.
- Investigate the properties of orthonormal bases and their significance in various mathematical contexts.
USEFUL FOR
Students and professionals in mathematics, particularly those studying linear algebra, as well as engineers and physicists who require a solid understanding of vector spaces and orthonormal bases.