Lines and planes

Vector spaces: Vector spaces and linear subspaces

Lines and planes

We now generalize the concepts line and plane to the context of a vector space.

Line
Let #\vec{p}# and #\vec{v}# be two vectors in a vector space and suppose #\vec{v}\neq \vec{0}#. Then the set of vectors
\[ \vec{p}+\lambda \cdot\vec{v} \quad\text{with }\ \lambda \in \mathbb{R} \]
is called a line in the vector space. The vector #\vec{p}# is called a support vector of the line and the vector #\vec{v}# a direction vector.

The description of the line as #\vec{p}+\lambda\cdot \vec{v}# is called a parametric representation of the line (with the parameter #\lambda#).

If the vector space is #\mathbb{E}^2# or #\mathbb{E}^3#, then the definitions concur with those of a parameterized line. Thus, the notion is indeed a generalization from the Euclidean plane and the Euclidean space to the case of a general vector space.

In the parametric representation only a sum and a scalar product of vectors appear. Therefore it is possible to give this more general definition.

Each vector of the line #l# with parametric representation
\[
\vec{p}+\lambda \cdot\vec{v}
\] can be a support vector of #l#: if #\alpha# is a fixed number, then it is easy to verify that the line #m# given by the parametric representation
\[
\left(\vec{p}+\alpha \cdot\vec{v}\right)+\lambda \cdot\vec{v}
\] is the same line as #l#. After all, by rewriting a vector #\left(\vec{p}+\alpha \cdot\vec{v}\right)+\lambda \cdot\vec{v}
# of #m# as #\vec{p}+\left(\alpha +\lambda\right)\cdot\vec{v}#, we see that the vectors of #m# also belong to #l#, and by rewriting #\vec{p}+\lambda \cdot \vec{v}# as #\left(\vec{p}+\alpha\cdot \vec{v}\right)+(\lambda-\alpha) \cdot\vec{v}#, we observe that each vector of #l# also belongs to #m#.

The direction vector of #l# is uniquely determined up to a scalar multiple.

Plane
Let #\vec{p}#, #\vec{v}#, #\vec{w}# be three vectors in a vector space such that #\vec{v}# is not a scalar multiple of #\vec{w}#, and #\vec{w}# is not a scalar multiple of #\vec{v}#. The collection of vectors
\[\vec{p} +\lambda\cdot \vec{v} +\mu\cdot \vec{w} \phantom{xxx}\text{with}\phantom{xxx} \lambda ,\mu \in \mathbb{R}\]
is called a plane in the vector space, with support vector #\vec{p}# and direction vectors #\vec{v}# and #\vec{w}#.

This description is called a parametric representation of the plane (with parameters #\lambda# and #\mu#).

If the vector space is #\mathbb{E}^3#, then the definitions correspond to the definitions given previously regarding a parameterized plane in space. Thus, we have indeed a generalization from the Euclidean plane and the Euclidean space to the case of a general vector space.

Later, we will formulate the conditions regarding the two direction vectors as: #\vec{v}# and #\vec{w}# are linearly independent.

Again, each vector of the plane can be a support vector.

The direction vectors can vary within the linear combinations of #\vec{v}# and #\vec{w}#, in such a way that the requirement that none of the two is a scalar multiple of the other, remains satisfied. This can be seen by use of arguments that are similar to the case of the line. The requirement has the effect that #\vec{v}\neq \vec{0}# and #\vec{w}\neq \vec{0}#.

Thus, lines and planes are sets of the form \[\vec{a}+W =\left\{\vec{a}+\vec{w}\mid \vec{w}\in W\right\}\] for a given vector #\vec{a}# and a linear subspace #W#. In the general case (of arbitrary subspaces #W# instead of just lines and planes, or: an arbitrary number of parameters), this collection is called an affine subspace.

Lines and planes have much to do with linear subspaces but are not always linear subspaces themselves:

Lines and planes through the origin

A line or plane is a linear subspace if and only if it contains #\vec{0}#. This is precisely the case if the support vector is a linear combination of the direction vectors.

Let #l# be a line. If #\vec{0}\in l#, we can use #\vec{0}# as support vector and describe the line by the parametric representation #\lambda\cdot \vec{v}# for a suitable vector #\vec{v}# which is not the zero vector. Since the sum #\lambda \cdot\vec{v}+\mu\cdot \vec{v}# can be written as #(\lambda +\mu)\cdot\vec{v}#, the sum of two vectors of #l# belongs to #l#. A scalar multiple #\mu\cdot (\lambda \cdot\vec{v})# also belongs to #l# since the vector can be written as #(\mu \cdot\lambda)\cdot \vec{v}#.

The proof for planes is similar.

These are special cases of a more general situation which will be discussed later.

In general, an affine subspace #\vec{a}+W#, where #\vec{a}# is a vector and #W# a linear subspace, is a linear subspace if and only if #\vec{a}# belongs to #W#.

Lines and planes are determined by two and three vectors, respectively:

Line and plane through a given set of points

Let #\vec{p}#, #\vec{q}#, and #\vec{r}# be three different vectors that are not all three on a line.

There is a unique line through #\vec{p}# and #\vec{q}#; it has parametric representation #\vec{p}+\lambda\cdot\left(\vec{q}-\vec{p}\right)#.
There is a unique plane that contains #\vec{p}#, #\vec{q}#, and #\vec{r}#; it has parametric representation #\vec{p} +\lambda\cdot\left( \vec{q}-\vec{p}\right) +\mu\cdot\left( \vec{r}-\vec{p}\right)#.

We are dealing indeed with a parametric representation of a line or a plane. Therefore, it is sufficient to verify that the indicated vectors belong to the line and plane, respectively:

For the line we find
- #\vec{p}# for #\lambda =0# and
- #\vec{q}# for #\lambda =1#.
For the plane we find
- #\vec{p}# for #\lambda =\mu =0# ,
- #\vec{q}# for #\lambda =1#, #\mu =0#, and
- #\vec{r}# for #\lambda =0#, #\mu =1#.

The subset \[ V=\left\{ \rv{x,y,z}\in\mathbb{R}^3 \mid 7 x+y-z =4\right \}\] of the vector space #\mathbb{R}^3# is a plane. Show this by giving a parametric representation of #V#.

# \rv{0,0,-4}+\lambda \cdot\rv{1,0,7}+\mu \cdot\rv{0,1,1}#

Take #x=\lambda# , #y=\mu#. Then #z=7\lambda +\mu -4#, so #V# consists of the vectors
\[\begin{array}{rcl} \rv{x,y,z}&=&\rv{\lambda,\mu,7\lambda +\mu -4}\\&=&\rv{0,0,-4}+\lambda \cdot\rv{1,0,7}+\mu \cdot\rv{0,1,1}\end{array} \]
We conclude that #V# is the plane in #\mathbb{R}^3# with position vector #\rv{0,0,-4}# and directional vectors #\rv{1,0,7}# and #\rv{0,1,1}#.

The equation #7 x+ y-z=4# is called the equation of the plane #V#. We have converted the equation to a parametric representation of the plane.

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