Vector representation of a line in ℝ²

Vectors: Lines and planes

Vector representation of a line in ℝ²

In the remainder of this chapter we discuss the application of vectors in geometry: in the plane \(\mathbb{R}^2\) and space \(\mathbb{R}^3\) we can introduce new linear algebra concepts more easily than in more complex vector spaces, where these concepts are also applicable. We start with a straight line in the plane

Vector representation of a straight line The vectors \(\vec{x}\) that point to the points on a straight line \(\ell\) can be generally described by a so-called vector representation (or parametric representation), that is, with a recipe of the following form: \[\ell : \,\vec{x} = \vec{u} + \lambda\cdot\vec{v}\] Here, the scalar \(\lambda\), also called in this context parameter, can be freely chosen and the only restrictions on the vectors \(\vec{u}\) and \(\vec{v}\) is that they are 2-dimensional and that \(\vec{v}\) is not the zero vector. Instead of \(\lambda\) you may use another letter, for instance \(t\) but in exercises we prefer to follow mathematical concention and use the Greek character.

The vector \(\vec{u}\) is a support vector of the line \(\ell\); this vector points to a certain point on the line. The vector \(\vec{v}\) is a so-called direction vector.

Line through the origin If \(\vec {u} = \vec{0} \) in the vector representation \(\vec{x} = \vec{u} + \lambda\cdot\vec{v}\) of the line \(\ell\), then the line passes through the origin. We usually omit this support vector in this case and write \(\ell : \,\vec{x} = \lambda\cdot\vec{v}\). Of course, you are free to choose a different support vector for this line, but then it is less clear at first sight that the line passes through the origin.

Support point Instead of support vector, we will also speak about support point. After all, it is all about a point in the plane that is the terminal point of the representative of the support vector with the origin as initial point.

The figure below illustrates the vector representation of a straight line. The support vector is coloured red, the direction vector is in green, and the scalar multiple of the direction vector is shifted to a black vector on the line. You can modify the support vector and direction vector by dragging their terminal points.

GeoGebra

Verify that after displacement of the support vector only the position of the line changes, but not its direction.

\(\phantom{x}\)

Look at the following examples to learn how to determine

a vector representation of a line through two points;
an equation of a line, in the form \(a\cdot x+b\cdot y=c\), when the vector representation of the line is given;
a vector representation of a line when the equation of the line is known.

The line \(\ell\) goes through \(\cv{-3\\3}\) and \(\cv{2\\5}\).
Give a vector representation of \(\ell \) with parameter \(\lambda\).

\(\displaystyle \cv{x\\y} = {}\)\(\cv{-3\\3} + \lambda\cdot \cv{5 \\ 2}\)
(There are more correct answers)
The terminal point of the vector \( \cv{-3\\3} \) lies on \(\ell\) and therefore it is a support vector. For the direction vector we choose the difference of the two given vectors.
For the direction vector we select the vector with initial point \(P\) and terminal point \(Q\), i.e., \(\vec{PQ}=\vec{OQ}-\vec{OP}\): \[\vec{PQ}=\cv{2\\5} -\cv{-3\\3} = \cv{2 +3 \\ 5 - 3} = \cv{5 \\ 2}\tiny.\] So, a possible vector representation is \[\cv{x\\y} = \cv{-3\\3} + \lambda \cv{5 \\ 2}\tiny.\]
More vector representations are possible, for example: \[ \cv{x\\y} = \cv{2\\5} + \lambda \cv{10 \\ 4}\tiny.\] Any vector on \(\ell\) can be used as a support vector and every non-zero multiple of the direction vector can be used as direction vector.

New example