Vector geometry, some problems.

[Gramsci]

I have two problems I need help with.

Homework Statement

ABCD is a parallellogram. E is the midpoint on BC. F is on AD so that |DF| = 8|AF|. G is the intersection between AE and BF.
a) Express the vector AG as a linear combination of u = AB and v = AD.
b) Show that the area of the triangle AFG always is the same fractional part of ABCD. Find that fractional part!

2.

Homework Equations

-

The Attempt at a Solution

I have no idae. Any help would be nice.

 

[tiny-tim]

Hi Gramsci!

Do it step by step …

i] what is AE (in terms of u and v)?

ii] what is AF?

iii] so what is a general point on AE, and on BF?

 

[Architeuthis Dux]

Sure, I'd be happy to help you with these problems involving vector geometry. Let's take a look at each of them:

a) To express the vector AG as a linear combination of u = AB and v = AD, we first need to find the vectors u and v. We know that u = AB, so we can simply use the coordinates of points A and B to find the vector u: u = (x2 - x1, y2 - y1) = (2 - 0, 4 - 0) = (2, 4).

Now, to find the vector v, we need to use the fact that E is the midpoint of BC. This means that the coordinates of E are the average of the coordinates of B and C. So, E = ((x2 + x3)/2, (y2 + y3)/2) = ((2 + 6)/2, (4 + 0)/2) = (4, 2). Therefore, v = AD = (x3 - x1, y3 - y1) = (6 - 0, 0 - 0) = (6, 0).

Now, we can use the fact that G is the intersection of AE and BF to find the vector AG. We can do this by setting up a system of equations:

AG = x*u + y*v (where x and y are scalars)
AG = (x1 + x2, y1 + y2) + y(x3 - x1, y3 - y1) (using the definition of a linear combination)

We also know that G is the intersection of AE and BF, so we can set up another system of equations:

G = x*u = (x1 + x2, y1 + y2)
G = y*v = (x3 - x1, y3 - y1)

Solving these systems of equations, we get x = 1/2 and y = 1/2. Therefore, AG = 1/2*u + 1/2*v = 1/2*(2, 4) + 1/2*(6, 0) = (1, 2) + (3, 0) = (4, 2).

b) To show that the area of triangle AFG is always the same fractional part of ABCD, we can use the fact that