Help with area formula using direction cosines

[davidwinth]

TL;DR Summary

Where does the area formula used in this video come from?

I am watching a youtube video and the guy is finding the area of triangles in the coordinate planes as shown in the image still below.

So the triangle ABC has area a and normal n with direction cosines {L,M,N}. He then says it is obvious from geometry that the area of triangle OAC is given as aL, the area of OBC is given as aN, and the area of triangle OAB is given by aM. This is not at all obvious to me. Isn't L just the cosine of the angle between the x-axis and the normal, and likewise for the others? Is there some theorem or something I am missing? How does this work?

Thanks!

 

[jedishrfu]

Can you post a link to the video?

What math course is this from?

It looks like they are applying vector cross-product and vector dot product notions to the normal vectors of each triangle.

Let's define three unit normal vectors:
- unit vector on the x axis is $\vec x$
- unit vector on the y axis is $\vec y$
- unit vector on the z axis is $\vec z$

Triangle ABC has an area of $a$ and a unit normal vector $\vec n$.
Triangle AOC is in the YZ plane and has a unit normal vector of $\vec x$

When you project ABC onto the YZ plane you get triangle AOC.

You do that projection via the inner (dot) product of the ABC normal vector $a \vec n$ with the unit vector $\vec x$ and recall that the inner product of $\vec n$ with $\vec x$ is the given direction cosine value $l$.

Similarly, for the other two triangles, AOB, whose unit normal vector is $\vec y$ and BOC, whose unit normal vector is $\vec z$

 

[davidwinth]

Hello,

The video is here:

It is not strictly a mathematics video. It is an engineering video which uses math so there is no further discussion of this point other than what I noted.

I do see that the areas are projections, and I know how dot products work for vectors as far as projecting one vector onto another, but what I am missing is a rule or theorem that says something like, "If we have a vector N that is oriented normal to a planer area A and the length of that vector is numerically equal to the area A, then the projection of that vector onto a unit vector U which is normal to another plane R is equal to the the area of the projection of A onto R." That's the part I am not understanding how to see. Is that just a general rule? Does it work if I project a vector representing a volume onto another volume's unit vector or something similar?

It seems to me that you kind of restated the fact without getting to why this works. Is this a theorem from analytic geometry or something like that? I'd love to see a proof but I don't even know where to look.

 

[A.T.]

but what I am missing is a rule or theorem that says something like, "If we have a vector N that is oriented normal to a planer area A and the length of that vector is numerically equal to the area A, then the projection of that vector onto a unit vector U which is normal to another plane R is equal to the the area of the projection of A onto R."

Such a projection scales the projected shape along one dimension by the cosine of the angle between the shape and plane. So it's area also changes by that factor.