> Given a vector x expressed in the standard basis, we change coordinates by applying

The set of lines in Euclidean 2-space is a vector space. There is an isomorphism into R^3: choose an origin, choose x,y axes, and then ax + by + c = 0 gives tuples in R^3.

The line is x, the tuple in R^3 is t but the wording uses x when it should use t (which is never introduced). The wording conflates vectors with coordinate vectors.