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)!

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

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