Enter all but one of the vectors and their norms into the calculator to determine the Gram Schmidt Orthonormalization. This calculator can also evaluate any of the vectors given the others are known.
Gram Schmidt Orthonormalization Formula
The following formula is used to calculate the Gram Schmidt Orthonormalization. For a set of vectors {v1, v2, …, vn}, the Gram Schmidt process generates an orthonormal set {u1, u2, …, un} using the following formulas:
u1 = v1 / ||v1||u2 = (v2 – (v2.u1)u1) / ||v2 – (v2.u1)u1||u3 = (v3 – (v3.u1)u1 – (v3.u2)u2) / ||v3 – (v3.u1)u1 – (v3.u2)u2||
Variables:
- vi is the i-th vector in the original set
- ui is the i-th vector in the orthonormal
- set||v|| denotes the norm (length) of vector v
- v.u denotes the dot product of vectors v and u
To calculate the Gram Schmidt Orthonormalization, start with the first vector, divide it by its norm to get the first orthonormal vector. For the next vector, subtract from it the projection of it onto all previously calculated orthonormal vectors, then divide by its norm. Repeat this process for all vectors in the original set.
What is a Gram Schmidt Orthonormalization?
Gram Schmidt Orthonormalization is a mathematical process used to convert a set of vectors into an orthogonal or orthonormal basis. This process is important in various fields such as physics and computer science, where it is used to simplify calculations involving vectors. The process involves taking a set of vectors, and through a series of calculations, transforming them into a set of orthogonal or orthonormal vectors, which are perpendicular to each other and have a magnitude (length) of one.
