Independent of Length

To form a 2D column matrix that is perpendicular to an other:

Swap the elements, and negate one of them.

This only works in 2D, however. It gives you one of an infinite number columns orthogonal to the given one. For example, all the following vectors are orthogonal to ( -5, 3)^T:

• ( 3, 5 )^T
• ( -3, -5)^T
• ( 1.5, 2.5)^T
• ( 6, 10)^T
• .... and so on

The reason this works is: If u is orthogonal to v, then u · v   =   0. So (ku) · v   =   k(u · v)  =   0, for any real number k. So there are an infinite number of vectors (ku) orthogonal to v.

Often one wishes to find a unit normal to a given vector. A unit normal to a given vector is a vector that:

1. Is orthogonal (normal) to the given vector.
2. Has a length of one.

Remember not to confuse the two ideas normalizing a vector (making a unit vector in the same direction as the vector), and computing a unit normal (making a unit vector in an orthogonal direction to a vector.)