Lines in Space

This section expects students to have familiarity with the notation and operations of vectors from . This section will develop the parametric form of a line and describe its advantages and drawbacks (like non-uniqueness of representations). The idea of parametric equations and their vector presentations are motivated throughout this section, as well as a reminder of local linearity from single variable calculus. is an activity that some instructors may have skipped over but offers a nice connection between the algebra of linear combinations and the construction of the vector equation of a line.

This section can be covered in a longer class period alongside Planes. Some faculty will omit the Preview Activity in this case and we would suggest that students can do this Preview Activity as a follow up exercise to verify the development of the equations of a plane.

This section includes extra randomized exercises to allow students to practice with a variety of ways that the direction and base point information can be given in both algebraic and geometric forms.

In single variable calculus, we learn that a differentiable function is locally linear. In other words, if we zoom in on the graph of a differentiable function around a point, the graph will look like the tangent line to the function at that point. In you can look at the graph of a curve and zoom in around a point on that curve to see that eventually the graph of the function will look linear. You can use the sliders at the top of change how zoomed in the plot is displayed or change the location on the curve that is being highlighted. You should visually verify at a several points that this function is differentiable because at any point shown, the graph is locally linear.

Recall that linear functions have the form \(f(x)=ax+b\). These functions play important roles in single variable calculus and are useful in approximating differentiable functions, in approximating roots of functions (Newton's Method), and approximating solutions to first-order differential equations (Euler's Method). In multivariable calculus, we will study curves in space and we will see that connection between local linearity and differentiability exists for curves in three (or more) dimensions. As we extend our study of functions to include those of two (or more!) variables in , we will see that a function is locally linear at a point if the surface defined by the function looks like a plane (the tangent plane) as we zoom in on the graph.

Because of their important role in understanding curves and surfaces in \(\R^3\), it is important for us to understand both lines and planes in space, as can both be thought of as generalizations of lines in two dimensions to three-dimensional space. In your study of one-variable functions, you probably worked with slope, direction, and measuring change using lines several times before you generalized those ideas for a curve in general. Similarly, we will want to understand how we talk about direction and measuring change along flat objects in \(\R^3\) (lines and planes) before we start generalizing to curved objects. In this section, we discuss how we can use vectors to describe lines in space. The next section will be dedicated to understanding planes.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

The way most people draw a line is more connected to the ideas in than slope or algebraic forms. Take a minute and draw a plot of the line described by \(x+y-1=0\). You should not continue reading until you have made a plot (by hand) of \(x+y-1=0\) that you would be proud to share with classmates.

You may have done some algebra or other calculation to get to slope-intercept form or another familiar algebraic structure, but when you went to draw the line you likely did the following steps:

1. \(y\)
2. Found a second point on your plot (perhaps using the slope to find the second point).
3. Drew a line segment from the first point to the second, then extended the line past the second point to the edge of the plot.
4. Returned to the first point and extended the line in the direction opposite of the second point.

In two-dimensional space, a non-vertical line is defined to be the set of points satisfying the equation \[\begin{aligned}\end{aligned}\] for some constants \(m\) and \(b\). The value of \(m\) (the slope) tells us how the dependent variable changes for every one unit increase in the independent variable, while the point \((0,b)\) is the \(y\)-intercept and anchors the line to a location on the \(y\)-axis. Alternatively, we can think of the slope as being related to the vector \(\langle 1, m \rangle\), which tells us the direction of the line, as shown on the left in Figure. Thus, we can identify a line in space by fixing a point \(P\) and a direction \(\vv\), as shown on the right. Since we also have vectors in \(\R^n\) to provide direction, this same idea of a point and a direction determining a line works in \(\R^n\) for any \(n\).

The vector \(\vv\) in is called a direction vector for the line. As we saw in Preview Activity, any vector parallel to \(\vv\) will have the form \(t \vv\) for some scalar \(t\). As a result, any vector emanating from the point \(P\) in a direction parallel to the vector \(\vv\) will be of the form \[\begin{aligned}\end{aligned}\] for some scalar \(t\). (Here we use \(O\) to denote the origin.)

To summarize, the terminal points of the vectors of the form in define a linear function \(\vr\) in space of the following form, which is valid in any dimension and gives some geometric insight into lines in higher dimensions.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

The vector form of a line, \(\vr(t) = \vr_0 + t\vv\) in Equation, describes a line as the set of terminal points of the vectors \(\vr(t)\). If \(\vr_0 = \langle x_0, y_0, z_0 \rangle\) and \(\vv = \langle a, b, c \rangle\), then we can view this vector form in terms of components and get the following: \[\begin{aligned}\vr(t) \amp= \langle x(t), y(t), z(t) \rangle \\ \amp= \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle \\ \amp= \langle x_0+at, y_0+bt, z_0+ct \rangle\end{aligned}\] Splitting this vector form into each component equation gives \[\begin{aligned}x(t) \amp= x_0 + at \amp y(t) \amp= y_0 + bt \amp z(t) \amp= z_0 + ct\end{aligned}\] These equations describe the coordinates of the points on the line separately where each is coordinate is a function of \(t\). The variable \(t\) represents an arbitrary scalar and is called a parameter. In particular, we use the following language.

The parametric equations for a line through the point \(P = (x_0, y_0, z_0)\) in the direction of the vector \(\vv = \langle a,b,c \rangle\) are \[\begin{aligned}x(t) \amp= x_0 + at \amp y(t) \amp= y_0 + bt \amp z(t) \amp= z_0 + ct\end{aligned}\]

Notice that there are many different parametric equations for the same line. For example, choosing another point \(P\) on the line or another (parallel) direction vector \(\vv\) produces another set of parametric equations. In many physical applications, it is useful to think of \(t\) as a time parameter and the parametric equations as telling us where we are on the line at each time. In this way, the parametric equations describe a particular way to walk along the line. There are, of course, many possible ways to walk along the same line, including in opposite directions and at different speeds. In mathematical terms, we say that the parameterization of a line is not unique.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.