Calculus: Analytic Geometry and Calculus, with Vectors

84 Vectors and geometry in three dimensions

21 Supposing that A, B, C are not all zero and that D2 0 D1, let ir1 and 1r2 be
the planes having the equations

(1)

(2)

Ax+By+Cz+D1 =0

Ax+By+Cz+D2=0.

Show that the planes have parallel normals and hence that the planes are parallel.

Show again that the planes must be parallel by showing that they have no point

of intersection; if the coordinates of a point P(x,y,z) satisfy (1), they certainly

cannot satisfy (2). Supposing that A and u are constants not both zero, show

that the equation

(3) A(14x+By+Cz+Dl) +μ(Ax+By +Cz+D2) = 0

is the equation of a plane parallel to trl and 1r2 unless A +.o = 0. Supposing

that Po(xo,yo,zo) is a given point, show that A and μ can be so determined that the

graph of (3) contains Po. Solution of last part: The graph of (3) will be a plane

containing Po if and only if A + u s 0 and

(4) A(Axo + Byo + Czo + D1) + μ(Axo + Byo + Czo + D2) = 0.

Since D1 0 D2, the coefficients of A and g are different numbers that we can call

E and F. We can then put (4) in the form

(5) AE+kF=0

and obtain a solution of our problem by setting A = F and ,t = -E because, in

this case, A + μ = F - E 0 and (4) holds.

22 Supposing that the graphs of the equations

(1) Alx + By + Clz + D1 = 0

(2) A2x + B2y + C2z + D2 = 0

are distinct (that is, different) planes 7r1 and ,r2 that intersect in a line L and that A

and p are constants not both zero, show that the equation

(3) A(Alx + Biy + C1z + D1) + A(-42x + B2y + C2z + D2) = 0

is the equation of a plane 7r containing the line L. Solution: It is clear that if

P(x,y,z) lies on L, then the coordinates of P satisfy both (1) and (2) and hence (3)

To prove that (3) is the equation of a plane, we must prove that the three

equations

AA1 + s42 = 0, AB1 + liB2 = 0, AC1 + 4aC2 = 0

cannot be simultaneously satisfied when A and μ are not both zero. This matter

is more delicate. If we suppose that the three equations are satisfied and A 0 0,

we find that

Al = (-μ/A)A2, B1 = (-,u/A)B2, C1 =(-u/)')C2

and obtain a contradiction of the hypothesis thatir1 and ire are not parallel.

The case μ 5x!5 0 is similar.

(^23) Using the hypotheses and equations of the preceding problem, show how to
determine A and μ such that (3) will be the equation ofa plane 7r containing L