algebraic geometry,projective equivalence

I am stuck in the following question.can anyone please help? We know that through any $n+3$ points in general position in $\mathbb P^n$(projective space of order n), passes a unique rational normal curve. Let $p_1,p_2,.....p_{n+3}$ be such points and let $q_i$ be the pullback of $p_i , i =1,2,...n$. Now we are asked to show that $(p_1,p_2,....p_{n+3})$ is projectively equivalent as an ordered set to another such collection $(p_1',...p_{n+3}')$ iff the corresponding ordered subsets $(q_1,....q_{n+3})$ and $(q_1',....q_{n+3}')$ in $\mathbb P^1$ are projectively equivalent that is iff the cross ratios $(q_1,q_2,q_3,q_i)=(q_1',q_2',q_3',q_{i}')$ for each $i = 4,....n+3$.

This is a problem from page $12$ of Harris's first course book.