How many elements are in the projective line $\mathbb{P}^{1}(k)$ if k is a finite field

Assume k is a finite field with n elements, how many elements are in the projective line $\mathbb{P}^{1}(k)$ and how do I work this out?

I know that an element of $\mathbb{P}^{1}(k)$ is represented by $[a, b]$, where $a, b \in k$, not both of the coordinates are 0, and two elements $[a, b]$ and $[c, d]$ are equal if for some $\lambda \in k^{*}$ we have $a=\lambda c, b=\lambda d$

However, I’m not sure how I can use this to work out the number of elements?

Likewise how would I advance this to work out the number of elements in $\mathbb{P}^{2}(k)$ where the elements are the triples [a,b,c] ?

Solution 1:

The elements are (in homogeneous coordinates):

$$(0,1),(1,1),\ldots, (q-1,1), (1,0),$$

where $k = \{0,1,\ldots,q-1\}$ has $q$ elements. So the number of elements is $q+1$.

In the first $q$ elements, the 2nd coordinate is normalized to $1$. In the last element, the 2nd coordinate is $0$ and the first coordinate is normalized to $1$.

Using normalization, the projective plane can be described similarly.