find orthogonal vectors

[Pulkit Varshney]

i have only one vector [x1 x2 x3]. it is 3 dimensional vector.
how can i find its other two orthogonal vectors in which all components must be non zero ?

 

[Shug]

You can use the Gram-Schmidt method:

https://en.wikipedia.org/wiki/Gram–Schmidt_process

Basically, select any 2 random vectors and apply the Gram-Schmidt process to get a set of 3 orthogonal vectors. There is probably an easier method, but that one is straightforward from what I can think of off the top of my head.

In truth there are infinite vectors that are orthogonal to that vector. These vectors lie in a plane, and any two orthogonal vectors in that plane will suffice.

 

[albbg]

Two vectors are orthogonal one each other if their dot product is zero, then I think you just have to find a vector (y1, y2, y3) so that:

x1*y1+x2*y2+x3*y3=0

As said by Shug, in the space there are infinite vector orthogonal to a given one.

If you don't have other costraints I think you can arbitrary assign a value to two variables among y1, y2 and y3 and find the third.

 

[Pulkit Varshney]

actually i am seeking for vectors which are mutually orthogonal, so in 3-d space there can be only 3 such vectors.
and i can not use gram-schmidt here because for that i have to have 3 (non-orthogonal) vectors, but i have only one.

 

[Shug]

actually i am seeking for vectors which are mutually orthogonal, so in 3-d space there can be only 3 such vectors.
and i can not use gram-schmidt here because for that i have to have 3 (non-orthogonal) vectors, but i have only one.

This is not correct... any two orthogonal vectors in the plane that is orthogonal to the vector will suffice. And you can use Gram-Schmidt; just pick any two random vectors and apply the Gram-Schmidt process to get 3 mutually orthogonal vectors.