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