# Projection of vector area onto $x$-$y$ plane

In our math class (in the context of introduction to vectors) we were told that the projection of the vector area $$S$$ onto the $$x$$-$$y$$ plane is equal to the dot product of the vector area $$S$$ and the unit vector in $$z$$. We don't prove it however and it doesn't seem obvious to me. Why is this geometrically true?

Why does multiplying the area by $$\cos(\text{angle})$$ give the correct projection? I also can't seem to find an answer online as most places I searched just assumed this formula to be true. Thanks in advance (and sorry if this is obvious)!

• decompose $S$ as the sum of the projection and the orthogonal part, that wll be parallel to $z$ – Exodd May 29 at 10:37
• I guess what is confusing me is why does the component of S parallel to z have the same area as the projection. – Peter May 29 at 10:42