This tool will allow you to calculate orthonormal bases of the subspace generated by vectors. It uses the popular Gram-Schmidt algorithm (2D Plan, 3D or 4D Space) in formal calculation.

Here, a set of non-zero vectors from a vector space is said to be orthogonal, given that the inner product between any two vectors in the set is equal to 0.

So, a set of vectors is said to be orthonormal if the set is orthogonal and if for any vector v in the set we have:

**Cv,vD = 1**

The original Gram-Schmidt theorem states that for any set of linearly independent vectors from a given vector space; it is always possible to generate an orthogonal set of variables with the same number of vectors as the original set.

The way to generate this set is done by constructing it from the original set of vectors by using Gram-Schmidt’s orthogonalization process:

If the projection of v onto u is mathematically given by- proj__u(*v*)=⟨*v*,*u*⟩⟨*u*,*u*⟩*u*, then they form a sequence of vectors as follows:

u__1=v__1

u__2=v__2−proj__u__1(v__2)

u__3=v__3−proj__u__1(v__3)−proj__u__2(v__3)

The sequence u__1,u__2,…u__k will be the required set of orthogonal vectors.