How does one divide by a vector when calculating pressure?

Question

Sir $P=F/A$ And since $F$ and $A$ both are vectors but $P$ is scalar. So doesn't it violates that "Division is NOT defined for vectors"?

Answer 1

Pressure is actually $$P=\frac{F_\bot}{A}$$ where $F_\bot$ is the force component perpendicular to the surface in question, and $A$ is the area of the surface.

Therefore, there is no "division by a vector" here. Certainly, the area vector is used in various areas of physics; this is not one of those areas (pun always intended).

I suppose if you wanted a definition based on vectors you can exploit the use of projections: $$P=\frac{\mathbf F\cdot\mathbf A}{||\mathbf A||^2}$$

since the area vector, by definition, is perpendicular to the surface in question.

Answer 2

division is not defined for vectors. However, it is obvious for vectors that are scalar multiples of each other. Just because we can apply division to a subset of vectors doesn't mean it's defined for the entire set