Why can't you add z=0 to all of them?
I know how to find the area of a triangle in three-space, say ABC: First I would find and then and then use the formula for the area of a parallelogram:
Finding the magnitude of that and dividing it by two (since the area of a triangle is half that of the area of a parallelogram), I would find the area of the triangle ABC.
But, how would I find the area of a triangle that does not have a Z component (is in two-space)? For example: Find the area of the triangle with vertices of (1, -2), (-1, 3), (2, 4). Since this is in two-space, I can't take the cross product of any two of these vectors to find the area. According to my book, the area of this triangle is and I have no idea how they computed that. Any help? I would appreciate learning a systematic approach to solving the areas of a triangle in two-space. Thanks in advance.
I did just that:
A(1, -2, 0), B(-1, 3, 0), and C(2, 4, 0)
The cross product of these two direction vectors:
Now to find the magnitude of the normal:
Since this is only the area of the parallelogram and the area of the triangle is half of that:
square units.
But in my textbook, the area of the triangle is or 8.5. So, did I compute the area incorrectly?
Minor mistake of mine, thanks for catching it. I usually use the dot product to confirm I computed the cross product correctly, but since z = 0 for all z, I foolishly assumed it was correct even though I was off.
Many thanks to TKHunny for giving me an invaluable tool to calculating problems like these.