Question

I have what I’m sure is a very simple problem but I can’t work it out.

I have two 2d vectors that form a line and I am looking to find the normals of this line. example:

vector 1 = ( -10 , 10 ) vector 2 = ( -10, -10 )

How do I calculate the normals for the line defined by these vectors?

Answer 1

It’s hard to tell which “normal” you want.

Do you mean out of the plane that the two vectors lie in? That’s the cross-product of the two. In this case it’s simple: (0, 0, 1) is the normal vector, because both lie in the xy-plane.

Do you mean one of the two normals in the plane for the line that runs from the head of vector 1 to the head of vector 2? All you need to do there is calculate the vector between them, exchange the values of the x- and y-components, and toggle the sign of either component.

In your case,

v2 - v1 = (-10-(-10))i + (-10-10)j = 0i - 20j

The normal vector is:

   n1 = 20i + 0j  (points in the positive x-direction)

   n2 = -20i + 0j (points in the negative x-direction)

Obviously you should normalize these to be unit vectors.

There are two vectors perpendicular to any line in a plane; they point in opposite directions.