1.5 Equations of lines and planes

Lines in ${\mathrm{R}}^{\mathrm{2}}$

There are several ways to write the equation of a straight line in ${ℝ}^2$. If the line is not parallel to the $y$-axis, then we can write: $y=mx+c$ where $m$ is the gradient of the line, and $c$ is the point where the line crosses the $y$-axis. If we know that the line passes through the point $(x_0,y_0)$ then we can write this equation in a slightly different form:

If $(x_1,y_1)$ is another point on the line then we can calculate $m$ as $\frac{y_1-y_0}{x_1-x_0}$.

We can also write the equation of the line in vector form: if $𝐯$ is a vector in the direction of the line, then the line is the set of all points of the form $(x_0,y_0)+\lambda 𝐯$ for $\lambda \in ℝ$. Once again, if $(x_1,y_1)$ is another point on the line then we can be more precise: in this case, we can take $𝐯=(x_1-x_0,y_1-y_0)$.

Let us look again at the equation $y=mx+c$. Rearranging, we obtain: $(xy)\cdot (-m\mathrm{\hspace{0.33em}\hspace{0.33em}1})=c$. Alternatively, $(x(y-c))\cdot (-m\mathrm{\hspace{0.33em}\hspace{0.33em}1})=0$. This can best be explained via a diagram:

The vector $𝐧=(-m\mathrm{\hspace{0.33em}\hspace{0.33em}1})$ is called a normal vector to the line $L$.

A normal vector to the line $ax+by=c$ is $(ab)$. To see this,

It may be useful to remember that $(xy)$ and $(y-x)$ are orthogonal vectors.

We saw above that in ${ℝ}^2$, a line can be described via an equation: $(xy)\cdot 𝐧=c$, where $𝐧$ is a normal vector to the line. What happens in ${ℝ}^3$? Let’s consider a specific example: let $𝐧=(\mathrm{0\hspace{0.33em}\hspace{0.33em}0\hspace{0.33em}\hspace{0.33em}1})$ and $c=0$. Then $(xyz)\cdot 𝐧=z$, so the set of vectors satisfying $(xyz)\cdot 𝐧=0$ is precisely the set of vectors of the form $(xy\mathrm{\hspace{0.33em}\hspace{0.33em}0})$. This is not a line but a plane, usually called the $x-y$ plane.

This demonstrates the following difference between ${ℝ}^2$ and ${ℝ}^3$:

- if $𝐮$ is a non-zero vector in ${ℝ}^2$, then the set of vectors orthogonal to $𝐮$ is a line,

- if $𝐮$ is a non-zero vector in ${ℝ}^3$, then the set of vectors orthogonal to $𝐮$ is a plane.

Following the same procedure as we used to find the equation of a line in ${ℝ}^2$, we can find the equation of a plane in ${ℝ}^3$.