### Vector calculus in plane and space: Straight Lines and Planes

### Straight lines and planes

From now on we assume that we have chosen an origin. The origin itself corresponds to the zero vector #\vec{0}#.

Parametric representation of a line

Let #\vec{v}# be a vector that is not equal to #\vec{0}#. The multiples #\vec{x}= \lambda \cdot\vec{v}# of #\vec{v}# run over the points/vectors of a *straight ** line* #\ell# through the origin. We call this line the line

**spanned**by #\vec{v}#.

If #\vec{a}# is a second vector, then the point #\vec{x} = \vec{a}+\lambda\cdot \vec{v}#, for , then #\lambda# varying over the real numbers, runs over the points of the line #m# through #\vec{a}# parallel to #\ell#. We call \[\ell : \,\vec{x} = \lambda\cdot\vec{v}\phantom{x} \hbox{ and }\phantom{x}m :\, \vec{x} = \vec{a} + \lambda \cdot\vec{v}\] a **parametric** or **vector representation** of the line #\ell# and #m#, resectively.

The vector #\vec{v}# is a so-called **direction vector** in both cases. The vector #\vec{a}# is a **particular vector** of the parametric representation of the line #m#. We call #\lambda# the **parameter**.

The line #\ell# is the special case of the general line #m# wherein #\vec{a}=\vec{0}#. The vector #\vec{0}# is a support vector of the line #\ell#.

The position and direction vector of a line are not unique:

- The vector #\vec{p} + \vec{v}# is located on the line #\ell# with parametric representation #\vec{x} = \vec{p} + \lambda \vec{v}#. Try entering #\lambda =1#. The vector #\vec{p}+\vec{v}# might as well act as a particular vector. The parametric representation \[\vec{x} = \vec{p}+\vec{v} + \mu \vec{v}\] is an equal parametric representation of the line #\ell#. For varying #\mu#, #\vec{p}+\vec{v} + \mu \vec{v}# in fact crosses the same vectors as #\vec{p} + \lambda \cdot\vec{v}# for varying #\lambda# (check). In fact, any vector can act as particular vector on #\ell#.
- In addition to #\vec{v}#,#2\vec{v}#, #-3\vec{v}#, #\pi \cdot \vec{v}# are direction vectors of #\ell#. The vectors of #\vec{p} + \mu\cdot (2\vec{v})# exactly cross the points of #\ell# at varying #\mu#.

Instead of *particular vector,* we will speak of **position point**. This is, after all, a point in the plane, which is the end point of the representative of the particular vector which has its starting point in the origin.

To summarize: in order for a parametric representation of a line, a particular vector and a direction vector are required.

Criteria for two vectors on a line through the origin

Two vectors can only be on a line through the origin if (at least) one of the two is a scalar multiple of the other.

The position vector #\rv{3,5}# is drawn in blue. The directional vector #\rv{-6,-5}# and the line #\ell# are drawn in black..

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