I need an explanation for the following graphic describing how to calculate the area of a triangle given three points(p1, p2, p3).
Intuitively, one can conclude that the area is equal to half the product of the base and the height to that base. My understanding is that, in this case, the base is (p2-p1), and the corresponding height is the second part of the equation.
What I am having a problem is with the second part describing what I think is a cross product of two vectors. How does the second part(cross product) form the appropriate height?
Thanks!
The cross product of two vectors in $3$-space may be interpreted as spanning a parallelogram, whose area is equal to the magnitude of the product. Half of this is the area of the triangle spanned by these two vectors.
As to why the area of this parallelogram represents the magnitude of the cross product, note that $a×b=|a||b|\sin\phi u,$ with $\phi$ being the angle between the vectors $a,b$ and $u$ is the unit vector in the direction of $a×b.$