## goldmahkot 2 years ago I was comparing the areas of a triangle that could be obtained via the conventional method, dot product, and cross product. Strangely, the latter did not agree with the former two. Consider two vectors, A <1,1,1> and B<1,1,0>. Both these vectors form a right angled triangle with an angle of 45deg between both vectors, a hypotenuse of SQRT(3), and length of SQRT(2).

1. goldmahkot

Now, the conventional method for finding the area of a triangle is (height X length)/2. In this case, it would be (SQRT(3)(sin45deg) X SQRT(2))/2. The answer obtained is 0.866.

2. goldmahkot

For the dot product, remember that we could find the area of a triangle with a "shortcut method" i.e. (|A'||B|sin(x))/2. Therefore, the area would be (SQRT(3)(sin45deg) X SQRT(2))/2 which gives an answer of 0.866.

3. goldmahkot

But when it comes to the cross product, things get a little strange.$A \times B = <-1,1,0>$To find its magnitude (which is equals to the area of a parallelogram): $\sqrt{(-1)^{2}+(1^{2}) + 0^{2}} = \sqrt{2}$To find the area of the triangle, we divide the answer above by 2: $\frac{ \sqrt2 }{2} = 0.7071$ The area obtained using the cross product is different from the other two methods!

4. Waynex

The area you calculated via the cross product is the correct one. The problem you have on the first two is that the angle between A and B is not pi/4. The angle would be pi/4 if both vectors were in the xy plane, but one vector is shift upward in angle, and that changes the angle between them. The angle is actual arccos[2/sqrt(6)], or aprox. 0.61 radians. The first equation would be sqrt(3) * sqrt(2) * sin(0.615...) * 0.5 = 0.707....

5. goldmahkot

Waynex, Thanks for you input on my question. You were right; I was assuming the angle between A and B to be pi/4 (a HUGE mistake). After recalculating the equations, all three areas agree with each other. I can now sleep in peace!