Compute the contraction of a 1-form with a vector field Question: Let $\alpha$ be the $1$-form on $\mathbb{R}^3$ given by $\alpha=zdy-ydz$ and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,y,-z)$. Compute $i_\mathbb{X}\alpha$
Answer: $i_{(0,y,-z)}(zdy-ydz)=i_{(0,y,-z)}(zdy)-i_{(0,y,-z)}(ydz)=zy-y(-z)=2yz.$
Is this answer correct? I feel that I must have done something wrong as I feel there should be more to the question.
I am not familiar with taking the contraction of a one form.
Can you just use the relation: $i_vf^{(j)}=v^j$?
Any feedback would be greatly appreciated.

$\beta$'s irrelevance notwithstanding, your computation of $\iota_{\mathbb X}\alpha$ is correct. I'm not sure what your final question is supposed to mean.
