Given a triangle vertices  F(-1,3/2,1), G(-1/2,-1/2,3/2), H(1/2,3/2,1/2).  Find the area of the triangle.

F = {-1, 1.5, 1};
G = {- .5, - .5, 1.5};
H = { .5, 1.5,  .5};
View3D[{Arrow[F, Blue], Graphics3D[{Blue, Text["F", 1.1 F]}], 
          Arrow[G, Blue], Graphics3D[{Blue, Text["G", 1.1 G]}], 
          Arrow[H, Blue], Graphics3D[{Blue, Text["H", 1.1 H]}], 
          Graphics3D[Polygon[{F, G, H}]]}];

Solution

First, we can use vector differences to find two position vectors equivalent to two of the sides of the triangle,  G - F  and  H - F.

View3D[{Arrow[F, Blue], Graphics3D[{Blue, Text["F", 1.1 F]}], 
          Arrow[G, Blue], Graphics3D[{Blue, Text["G", 1.1 G]}], 
          Arrow[G - F, Red], 
          Graphics3D[{Red, Text["G-F", 1.1 (G - F)]}], 
          Arrow[G - F, Tail -> F, Pink], 
          Graphics3D[{Pink, Text["G-F", 1.1 F +  .5 (G - F)]}], 
          Arrow[H, Blue], Graphics3D[{Blue, Text["H", 1.1 H -  .1 G]}], 
          Arrow[H - F, Red], 
          Graphics3D[{Red, Text["H-F", 1.1 (H - F)]}], 
          Arrow[H - F, Tail -> F, Pink], 
          Graphics3D[{Pink, 
              Text["H-F", 1.2 H -  .4 (H - F)]}]}];

The area of the parallelogram with sides  G - F  and  H - F  is the norm (or length) of the cross product,

P = (G - F)\[Cross](H - F);
View3D[{Arrow[P, Brown], 
          Graphics3D[{Brown, Text["P=(G-F)x(H-F)", 1.1 P]}], 
          Arrow[P, Tail -> F, Sienna], Arrow[F, Blue], Arrow[G, Blue], 
          Arrow[G - F, Red], 
          Graphics3D[{Red, Text["G-F", 1.1 (G - F)]}], 
          Arrow[G - F, Tail -> F, Pink], Arrow[H, Blue], 
          Arrow[H - F, Red], 
          Graphics3D[{Red, Text["H-F", 1.1 (H - F)]}], 
          Arrow[H - F, Tail -> F, Pink], Graphics3D[Polygon[{F, G, H}]], 
          Graphics3D[Line[{G - F, H - F}]], 
          Graphics3D[
            Polygon[{{0, 0, 0}, G - F, G + H - 2  F, 
                H - F, {0, 0, 0}}]]}];

The area of the triangle is (Sqrt[11])/2, half the area of the parallelogram, because the parallelogram is made of two triangles with equal bases and heights.