There's an example on "Elementary Geometry of Differentiable Curves:An Undergraduate Introduction" by Gibson on page 9 which is:
"Example 1.10 Let be a non-zero vector, and let be a line with equation . By a direction for the line we mean any
vector orthogonal to : it is an example of a 'tangent' vector.(Chapter 2) Thus we could take . Alternatively, we could choose
any two distinct points on the line, and take : the relation then follows from ,
on subtraction. On this basis we say that two lines , are orthogonal when the corresponding 'normal' vectors
, are orthogonal, or equivalently the corresponding 'tangent' vectors are orthogonal:
either way, the condition is that ."
My problem is: How did the author picked and to be normal vectors from the two equations?
Can anyone kindly tell me how and why are these normal vectors? And also how did the author(I know he's is 100% right) find out that and
are orthogonal? I can't seem to see the big picture.