Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences

Lines and planes are perhaps the simplest of curves and surfaces in three dimensional space. They also will prove important as we seek to understand more complicated curves and surfaces.

You may recall that the equation of a line in two dimensions is \(ax+by=c\text{;}\) it is reasonable to expect that a line in three dimensions is given by \(ax + by +cz = d\text{.}\) However it turns out that this is the equation of a plane. We will turn our attention to a study of planes and return to consider lines later in this section.

A plane does not have an obvious “direction” as does a line. It is possible to associate a plane with a direction in a very useful way, however: there are exactly two directions perpendicular to a plane. Any vector with one of these two directions is called normal to the plane. While there are many normal vectors to a given plane, they are all parallel or anti-parallel to each other.

Suppose two points \((v_1,v_2,v_3)\) and \((w_1,w_2,w_3)\) are in a plane; then the vector \(\langle w_1-v_1,w_2-v_2,w_3-v_3\rangle\) is parallel to the plane. In particular, if this vector is placed with its tail at \((v_1,v_2,v_3)\) then its head is at \((w_1,w_2,w_3)\) and it lies in the plane. As a result, any vector perpendicular to the plane is perpendicular to \(\langle w_1-v_1,w_2-v_2,w_3-v_3\rangle\text{.}\) In fact, it is easy to see that the plane consists of precisely those points \((w_1,w_2,w_3)\) for which \(\langle w_1-v_1,w_2-v_2,w_3-v_3\rangle\) is perpendicular to a normal to the plane, as indicated in Figure 6.11. Turning this around, suppose we know that \(\langle a,b,c\rangle\) is normal to a plane containing the point \((v_1,v_2,v_3)\text{.}\) Then \((x,y,z)\) is in the plane if and only if \(\langle a,b,c\rangle\) is perpendicular to \(\langle x-v_1,y-v_2,z-v_3\rangle\text{.}\) In turn, we know that this is true precisely when \(\langle a,b,c\rangle\cdot\langle x-v_1,y-v_2,z-v_3\rangle=0\text{.}\) That is, \((x,y,z)\) is in the plane if and only if

Working backwards, note that if \((x,y,z)\) is a point satisfying \(ax+by+cz=d\) then

Namely, \(\langle a,b,c\rangle\) is perpendicular to the vector with tail at \((d/a,0,0)\) and head at \((x,y,z)\text{.}\) This means that the points \((x,y,z)\) that satisfy the equation \(ax+by+cz=d\) form a plane perpendicular to \(\langle a,b,c\rangle\text{.}\) (This doesn't work if \(a=0\text{,}\) but in that case we can use \(b\) or \(c\) in the role of \(a\text{.}\) That is, either \(a(x-0)+b(y-d/b)+c(z-0)=0\) or \(a(x-0)+b(y-0)+c(z-d/c)=0\text{.}\))

Thus, given a vector \(\langle a,b,c\rangle\) we know that all planes perpendicular to this vector have the form \(ax+by+cz=d\text{,}\) and any surface of this form is a plane perpendicular to \(\langle a,b,c\rangle\text{.}\)

We will frequently need to find an equation for a plane given certain information about the plane. While there may occasionally be slightly shorter ways to get to the desired result, it is always possible, and usually advisable, to use the given information to find a normal to the plane and a point on the plane, and then to find the equation as above.

We will now turn our attention to a study of lines. Unfortunately, it turns out to be quite inconvenient to represent a typical line with a single equation; we need to approach lines in a different way.

Unlike a plane, a line in three dimensions does have an obvious direction, namely, the direction of any vector parallel to it. In fact a line can be defined and uniquely identified by providing one point on the line and a vector parallel to the line (in one of two possible directions). That is, the line consists of exactly those points we can reach by starting at the point and going for some distance in the direction of the vector. Let's see how we can translate this into more mathematical language.

Suppose a line contains the point \((v_1,v_2,v_3)\) and is parallel to the vector \(\langle a,b,c\rangle\text{.}\) If we place the vector \(\langle v_1,v_2,v_3\rangle\) with its tail at the origin and its head at \((v_1,v_2,v_3)\text{,}\) and if we place the vector \(\langle a,b,c\rangle\) with its tail at \((v_1,v_2,v_3)\text{,}\) then the head of \(\langle a,b,c\rangle\) is at a point on the line. We can get to any point on the line by doing the same thing, except using \(t\langle a,b,c\rangle\) in place of \(\langle a,b,c\rangle\text{,}\) where \(t\) is some real number. Because of the way vector addition works, the point at the head of the vector \(t\langle a,b,c\rangle\) is the point at the head of the vector \(\langle v_1,v_2,v_3\rangle+t\langle a,b,c\rangle\text{,}\) namely \((v_1+ta,v_2+tb,v_3+tc)\text{;}\) see Figure 6.12.

In other words, as \(t\) runs through all possible real values, the vector \(\langle v_1,v_2,v_3\rangle+t\langle a,b,c\rangle\) points to every point on the line when its tail is placed at the origin. It is occasionally useful to use this form of a line even in two dimensions; a vector form for a line in the \(x\)-\(y\)-plane is \(\langle v_1,v_2\rangle+t\langle a,b\rangle\text{,}\) which is the same as \(\langle v_1,v_2,0\rangle+t\langle a,b,0\rangle\text{.}\)

Another common way to write this is as a set of parametric equations: