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.

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.

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 $V=\left\{ \rv{x,y,z}\in\mathbb{R}^3 \mid 2 x+7 y-z =6\right \}$ of the vector space $\mathbb{R}^3$ is a plane. Show this by giving a parametric representation of $V$ .

$\rv{0,0,-6}+\lambda \cdot\rv{1,0,2}+\mu \cdot\rv{0,1,7}$

Take $x=\lambda$ , $y=\mu$ . Then $z=2\lambda +7 \mu -6$ , so $V$ consists of the vectors
$\begin{array}{rcl} \rv{x,y,z}&=&\rv{\lambda,\mu,2\lambda +7\mu -6}\\&=&\rv{0,0,-6}+\lambda \cdot\rv{1,0,2}+\mu \cdot\rv{0,1,7}\end{array}$
We conclude that $V$ is the plane in $\mathbb{R}^3$ with position vector $\rv{0,0,-6}$ and directional vectors $\rv{1,0,2}$ and $\rv{0,1,7}$ .

The equation $2 x+ 7 y-z=6$ is called the equation of the plane $V$ . We have converted the equation to a parametric representation of the plane.

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