In this section, we review the different ways we can represent lines and planes, including parametric representations.

Representations of Lines

When you think of describing a line algebraically, you might think of the standard form $$y = mx + b,$$ where $m$ is the slope and $b$ is the $y$-intercept. This is often called slope-intercept form.

In addition to slope-intercept form, there are several other ways to represent lines. For example, you may remember using point-slope form in single variable calculus. We can describe a line of slope $m$ going through a point $(x_0, y_0)$ with the equation $$y-y_0 = m(x-x_0).$$ It’s important to note that there are many different possible choices for the point $(x_0,y_0)$. Because of this, unlike slope-intercept form, point-slope form does not give a unique representation of a line.

In linear algebra, we saw that we could parametrize a line using a vector $\vec {v}=(v_1,v_2)$ giving the direction of the line, and a point $(x_0, y_0)$ that the line passes through. We parametrize the line as

Note that this representation works a bit differently from the previous two representations. In slope-intercept form and point-slope form, the line was the set of points $(x,y)$ satisfying the given equation. However, in the parametrization, we plug in values for the parameter $t$ in order to get points on the line.

Unlike slope-intercept form and point-slope form, the parametrization of a line can easily be generalized to three or more dimensions. That is, a line in $\mathbb {R}^n$ through the point $\vec {a}$ and in the direction of the vector $\vec {v}$ can be parametrized as $$\vec {x}(t) = \vec {v}t + \vec {a},$$ for $t\in \mathbb {R}$.

If we would like to describe a line in higher dimensions using equations (rather than a parametrization), we would need more than one equation. For example, in $\mathbb {R}^3$, we would require two equations to determine a line.

Representations of Planes

We also have multiple ways to represent planes. Here, we’ll focus on planes in $\mathbb {R}^3$.

Recall that a plane can be determined by two vectors (giving the “direction” of the plane) and a point that the plane passes through. We can use this to give a parametrization for the plane through the point $\vec {a}$ and parallel to the vectors $\vec {v}$ and $\vec {w}$: $$\vec {x}(s,t) = \vec {v}s+\vec {w}t + \vec {a},$$ for $s$ and $t$ in $\mathbb {R}$. Note that we require two parameters for the parametrization of the plane.

We can also describe a plane using a single linear equation in $x$, $y$, and $z$. For example, $$2x+4y-z=9$$ defines a plane. A standard way to do this is using a point on the plane and a normal vector to the plane. Recall that a normal vector is perpendicular to every vector in the plane. If $\vec {n}=(n_1,n_2,n_3)$ is a normal vector to a plane passing through the point $\vec {a}=(a_1,a_2,a_3)$, the plane is defined by the equation $$\vec {n}\cdot (\vec {x}-\vec {a})=0.$$ This can be rewritten as $$n_1(x-a_1)+n_2(y-a_2)+n_3(z-a_3)=0.$$