Find the perpendicular vector .

Question

Suppose we have two vectors A and B. These two vectors are at some angle to each other. I need to find a third vector which is perpendicular to vector A keeping vector B in mind?

Answer 1

Assuming that vectors $A$ and $B$ are not parallel (or antiparallel), they form a plane. There are two perpendicular vectors to $A$ that you can create with this information. The first one is using the cross product: $$C=A\times B$$This is a vector perpendicular to the plane of $A$ and $B$, therefore perpendicular to both. The other option is to create a vector in the $AB$ plane, perpendicular to $A$ using Gram-Schmidt procedure: $$D=B-\frac{A\cdot B}{|A|^2}A$$ Now any vector in the plane of $C$ and $D$, which can be written as a linear combination $\alpha C+\beta D$ is perpendicular to $A$