Description of $(T\Bbb{CP}^1)^\perp$ Is there a nice "concrete" description (i.e., coordinates) of the normal bundle of $\Bbb{CP}^1$ when is considered as a submanifold of $\Bbb{CP}^n$? Or, at least, $\Bbb{CP}^2$? 
 A: $\newcommand{\Cpx}{\mathbf{C}}\newcommand{\CP}{\mathbf{CP}}$Yes, it's a direct sum of $(n - 1)$ copies of the hyperplane bundle $\mathcal{O}_{\CP^{1}}(1)$.
In fact, the total space of this bundle embeds holomorphically in $\CP^{n}$; the image is the complement of a linear subspace of dimension $(n - 2)$ disjoint from $\CP^{1}$. This is particularly easy to see in $\CP^{2}$: Pick a point not in $\CP^{1}$, and project away from it. The fibres of the projection are complex lines transverse to $\CP^{1}$.
In $\CP^{n}$ with $n \geq 2$, project away from a skew $(n - 2)$-dimensional subspace $P$; that is, if $x$ is a point of $\CP^{n} \setminus P$, the hyperplane containing $x$ and $P$ intersects $\CP^{1}$ at a unique point $\pi(x)$; the mapping $\pi:\CP^{n} \setminus P \to \CP^{1}$ is the projection of the normal bundle.

Edit to add requested details:
Geometrically, choose projectively independent points $(a_{j})_{j=1}^{n-1}$ in the $(n - 2)$-dimensional subspace $P$, i.e., $(n - 1)$ points represented by some basis of the $(n - 1)$-dimensional vector subspace of $\Cpx^{n+1}$ whose projectivization is $P$. For each point $b$ of $\CP^{1}$, the $(n - 1)$ lines $\ell_{j}(b) = \overline{a_{j}b}$ span the projective hyperplane containing $P$ and $b$, whose intersection with $\CP^{n} \setminus P$ is the fibre $\pi^{-1}(b)$. As $b$ varies over $\CP^{1}$, the lines $\ell_{j}(b)$ sweep out the projective plane containing $\CP^{1}$ and $a_{j}$, the total space of a hyperplane bundle over $\CP^{1}$ (with the point $a_{j}$ at infinity).
Algebraically, after a linear change of coordinates we may assume (as in my second comment below) that $[z_{0}: z_{1}: \dots: z_{n}]$ are homogeneous coordinates on $\CP^{n}$, that $\CP^{1}$ is defined by $z_{j} =0$ for $2 \leq j \leq n$, and that $P$ is defined by $z_{0} = z_{1} = 0$. Projection away from $P$ is then the coordinate projection
$$
\pi\bigl([z_{0}: z_{1}: z_{2}: \dots: z_{n}]\bigr) = [z_{0}: z_{1}: 0: \dots: 0].
$$
The normal bundle of $\CP^{1}$ in $\CP^{n}$ is the direct sum of the $z_{j}$-coordinate axes for $2 \leq j \leq n$, each of which is a hyperplane bundle over $\CP^{1}$. (In this second description, $a_{j}$ is the point of $\CP^{n}$ corresponding to the $j$th standard basis vector.)
