I'm working through Bachman's "A Geometric Approach to Differential Forms" and need a sanity check on one of the exercises. In Chapter 1, Exercise 2.2 it ask,...
Find a 1-form which projects vectors onto the line $dy = 2dx$ and scales by a factor of 2.
So my attempt at a solution is to find a vector on the line y = 2x that is 2 times longer than the unit vector and I get$<\frac{4}{\sqrt{5}},\frac{2}{\sqrt{5}}>$. Is this what the question is asking for? Is there something I'm missing? I tried to find a unit vector on that line and found $<\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}}>$ and feel pretty confident about that part. I figured the scaling should just be multiplying that vector by the scalar $2$. It's been a while since vector calculus and I haven't encountered a problem in a while that was asked quite like this one. Any help is appreciated!