Question

let f(t) be the area of the parallelogram with vertices v=(1,2t), w=(2t,1), (0,0) and v+w=(2t+1,2t+1). find a formula for f(t)

Answer 1

emt16, you can make a plot of this thing like I have attached.

Answer 2

There are a couple of ways of going about it. The easiest is to use the fact that the magnitude of the vector cross product gives the area of a parallelogram. Another way is to find the equation of the line for the base, say, and then use the slope of that base line to determine the slope of a line that is perpendicular to that line and cuts the point, (1,2t), say.

I think it's easier to use vectors. The vector of AB is given by

B - A => (1-0,2t-10) = (1,2t)

The vector AD is

D - A => (2t-0,1-0) = (2t,1)

Answer 3

The cross product is then

AB x AD = (4t^2-1) in the direction perpendicular to the x-y plane.

The area is the magnitude of this vector, which is

f(t) = 4t^2-1

Answer 4

Are vectors your thing or not?

Answer 5

ok. i can see now how it works. vectors aren't my strongest point but this helps a lot. thanks

Answer 6

Good. It's easier than using coordinate geometry. You'd have to find equations of lines and distances and everything...would take forever.