write your ideas down和write down your ideas两个都对。
down是一个副词，这是一个由“动词+副词”构成的短语动词，名词作宾语时既可放在后面，也可放在中间；如果是代词作宾语，只能够放在动词和副词之间，不能放在副词后面。再如pick it up;pick the book up,pick up the book都可以。
如果是“动词+介词”构成的'短语，无论名词还是代词作宾语，都应放在介词的后面，如look after your brother，look after him，而不能放在介词的前面。
write down例句：
After you've read it, please write down a few brief comments.
看完请签个意见。
Write down (in order of priority) the qualities you'd like to have.
（按照优先次序）写下你想具有的品质。
Write down the sort of thing you would like to do, given the opportunity.
写下有机会的话你想做的事情。