Thread: Linear combination and a proof for the area.

http://dl.dropbox.com/u/25711584/algebra.jpg





* express as a linear combination of the vectors u= and v= .

* Show that the triangle AFG always constitutes the same fraction of the area of the Parallelogram ABCD.


i need help here. On the first question i have tryed to express the vector AG with other vectors to finally come to an linear combination of u and v. But i get problems whit any vector that ends at G. I suppose i should use the symmetry of the triangles somehow but dont really know how?


the other question i need some tips to get started.

Thanks.

~ (You know why?)
and

Now this should be easy.

Originally Posted by mechaniac

http://dl.dropbox.com/u/25711584/algebra.jpg





* express as a linear combination of the vectors u= and v= .

* Show that the triangle AFG always constitutes the same fraction of the area of the Parallelogram ABCD.


i need help here. On the first question i have tryed to express the vector AG with other vectors to finally come to an linear combination of u and v. But i get problems whit any vector that ends at G. I suppose i should use the symmetry of the triangels somehow but dont really know how?


the other question i need some tips to get started.

Thanks.

1. According to your sketch:





2.



3. Since you'll get:



Expand both sides of the equation and compare the co-efficient of equal vectors:



Solve this system of equations for t and k.

4. Plug in the value for t or k into the appropriate equation to get the vector

Thanks for the help! i used determinants to show the area problem so dont need help on that