## shelovespiano 3 years ago Let u=(4,0,1), v=(5,-1,0), and w=(-3,1,-2). Find the area of the triangle determined by u and v.

1. slaaibak

I think the area for the parallelogram would be the length of the cross product. |u x v|

2. slaaibak

Therefore, the area of the triangle would be the half of }u x v|

3. shelovespiano

Thanks!

4. shelovespiano

Could you write it out? I'm a little confused, I guess.

5. amistre64

what have you some up with in the cross product so far?

6. shelovespiano

I have (20, 0, 0) but I feel like I've done something wrong... I'm not very good with vectors.

7. amistre64

$\begin{vmatrix}X&Y&Z\\U_x&U_y&U_z\\V_x&V_y&V_z\end{vmatrix}\to X\begin{vmatrix}\\U_y&U_z\\V_y&V_z\end{vmatrix} -Y\begin{vmatrix}\\U_x&U_z\\V_x&V_z\end{vmatrix} +Z\begin{vmatrix}\\U_x&U_y\\V_x&V_y\end{vmatrix}$

8. amistre64

thats the "formal" confusion for it: me, i just do this x 4 5 x = 1 y 0 -1 y = 5 z 1 0 z = -4

9. shelovespiano

How does what you just did work?

10. amistre64

so, the length of the vector <1,5,-4> its the same process as the fancy typing; but in a vertical format that I can keep track of better

11. amistre64

turn the matrix on its side pretty much and expand down the first column; taking the determinant of each sub matrix along the way

12. amistre64

<1,5,-4> ^2 = 1 + 25 + 16 = sqrt(42) sqrt(42)/2 = area triangle

13. shelovespiano

thank you!!

14. amistre64

there are other ways to do it if you cant cross that well :)

15. amistre64

but the cross bypasses any trig you might not know

16. shelovespiano

Ok. I think this works pretty well for me:P