Which vector do you think is best for designing a plane? And by ‘best’ I mean ‘easiest to correct for scaling and invert’, or ‘easiest to correct for scaling inversions’? Orthogonal is always the best ‘easy to correct for’ option. Parallel, however, has a number of important disadvantages, particularly in this case of engineering a two-dimensional plane from 3D space.

Using these considerations you can decide on either your orthonormal or parallel vector bases and then the vector is automatically orthogonal. In cases where you know the projection of the vector you are projecting along the vector, if the basis vectors are orthogonal then there is no projection, or you can see them as parallel.

Knowing how to correctly solve for orthogonal vectors, for example using the Pythagorean Theorem and the dot product, makes creating parallel vectors much easier and will reduce stress from the dot product, which is the standard method for finding cosine similarity between two vectors.Using this process, you can determine if two vectors are orthogonal, or whether they are parallel.

This article on Vector Algebra uses the Pythagorean Theorem and the dot Product to determine orthogonality. New blog: Vector Algebra will help you understand the dot product.

The Pythagorean Theorem and the Dot Product will help you find orthogonal and parallel vectors. Using the Pythagorean Theorem and the dot Product will allow you to determine if two vectors are orthogonal or parallel.