Calculus: Analytic Geometry and Calculus, with Vectors

80 Vectors and geometry in three dimensions

where A (lambda) and μ (mu) are constants. Conversely, if A and μ are
not both 0, then (2.45) and (2.451) are equations of a plane containing L.
To see this, we notice that they have the form Ax + By + Cz + D = 0,
where A and B are not both 0, and that they are satisfied when x = xi,
y = yi, z = zi and when x = x2, y = y2, z = z2. Now we can solve
problems. Suppose we want to find the equation of the plane ui which
contains Pi and P2 and also a third point Pa(xa,ya,za) not on the line P1P2.
Our answer will be (2.45) or (2.451), and A and is are determined such
that they are not both zero and the formulas hold when x = x3, y = ya,
z = za. If the coefficient of A in (2.45) is zero, we can take A = I and
μ = 0; otherwise, we can set μ = 1 and solve for X. Suppose next that
we want to find the equation of the plane it which contains Pi and P2 and
is perpendicular to a given plane ir'. Our answer will be (2.451) when A
and i are determined such that they are not both 0 and a normal to x
is perpendicular to a normal to a'. Supposing that the equation of x is

Ax+Bv+Cz+D=0,

we find that the normals are perpendicular when

(2.46) aA +

;1B (a + μ)C

x2-X1 Y2-y1 Z2-z, = 0.

It is possible to find values of A and μ which satisfy this equation.

Let d be the distance from a given point Pi(xi,yi,z,) to a given plane x

having the equation

(2.47) Ax+By+Cz+D=0.

One way to find d is to find the point Po where the line through Pi per-

pendicular to v intersects it and then find

the distance from Po to Pi. Whether this

method is tedious or not can be a matter of

opinion, but it is quite lengthy even when

A, B, C, D, x1, y1, zi are given to be nice little

integers. With the aid of vectors, we can very

PA=1.y1.t1) easily find d in terms of A, B, C, D, xi, yi, zi.

Figure 2.471 Let P(x,y,z) be any point in ar, and let n be
a unit normal to ir. Then, as we can see with
the aid of the schematic Figure 2.471 in which Po and P are on ar and
PiPo is a normal to ir,

(2.472) d = I;PP,I cos 0, =

But

AI+Bj+Ck

n

__

,%/,42+ B2 +

C21 PP, = (xi - x)i + (y1 - y)J + (z1 - z)k.