Which of the following equations give alternate parameterizations of the line L parameterized by: r(t)=(1+2t)i +(2+2t)j -(1+4t)k?
a. -(1+t)i-t*j+(3+2t)k
b, (3-2t)i+(2-2t)j+(3-4t)k
c. (2+3t)i+(1+3t)j+(1-6t)k
We are working on the paramaterization of lines. We need to find which one of the the three choices is equal to the given equation. I set the components equal to each other to see if they satisfy the equations, and found the answer to be A, but i'm unsure if that is what I need to do.
You can split your given vector like this to make the role of $t$ obvious: $$r(t)=(i+2j-k)+(2i+2j-4k)t$$ So your line $L$ goes through the point $i+2j-k$ and is parallel (or antiparallel) to the vector $2i+2j-4k$.
Just look at each of your possible answers and check these conditions. The easiest way to check the parallel condition is to split the possible answer in the same way I split $r(t)$ and see if the second part (multiplied by $t$) is a multiple of $2i+2j-4k$.