How do I use cross products to find the area of the quadrilateral in the $$-plane defined by $(0,0), (1,−1), (3,1)$ and $(2,8)$?

Question

How do I use cross products to find the area of the quadrilateral in the $$-plane defined by $(0,0), (1,−1), (3,1)$ and $(2,8)$?

So what I first do is find two vectors. Gonna use (0,0) as the starting point cause that's easier.

My two vectors: $\left\langle 2,8,0\right\rangle$ and $\left\langle 1,-1,0\right\rangle$

Now I calculate the cross product and get: $\left\langle 0,0,10\right\rangle$

Now I find the magnitude and get 10. Divide 10 by 2 and get 5 as my area. Now, that definitely makes no sense and it isn't the correct answer.

What am I doing wrong?

Answer 1

That formula is for the area of a parallelogram.

Your quadrilateral is not a parallelogram.

You could divide it into two triangles

and then find the area of each triangle by taking half of the magnitude of the cross-product

and then add the areas together.

Answer 2

Take $\frac12$ of the cross product of the diagonal vectors $(3,1)$ and $(1,9)$. $$\operatorname{Area}=\frac12 d_1 d_2 \sin (\widehat{d_1,d_2})$$