How do we construct an Orthonormal Basis from a Single Vector a?

How do we construct an Orthonormal Basis from a Single Vector a?



By using Cross Product, we can find a new basis denoted as u, v, and w.

1. Make w a unit vector in the direction of a:
w = a / ||a||

2. Then choose any vector t not collinear with w, and use the Cross Product to build a unit vector u that is perpendicular to w:
u = t x w / ||t x w||

If t is collinear with w then the denominator will vanish. A simple procedure to find a vector sufficiently different from w is to start with a t equal to w and change the smallest magnitude component of t to 1.

3. v = w x u

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