I was watching a lecture. In that lecture, professor drew the following picture. He was saying, if u is a normal/perpendicular vector and V is a vector that's not perpendicular to the line, then the dot product of this two vector will be the distance/width between the two lines. Can any body explain it to me intuitively?
Take the definition of dot product as $$a\cdot b=|a||b|\cos \theta,$$ where $\theta$ is the angle between $a$ and $b$.
If $\theta$ is the angle between the vectors, the required distance is $|V|\cos\theta=V\cdot u$, by definition of dot product, and the fact that $|u|=1$