3 Different and not parallel planes

[Vigorous]

Homework Statement

Suppose we know that when the three planes P1, P2 and P3 in R3 intersect in pairs, we get three lines L1, L2, and L3 which are distinct and parallel.
a) Sketch a picture of this situation.

b) Show that the three normals to P1, P2 and P3 all lie in one plane, using a geometric argument.

c) Show that the three normals to P1, P2 and P3 all lie in one plane, using an algebraic argument. (Note that the three planes clearly do not all intersect at one point.)

Relevant Equations

1) Dot product
2) Cross product

b) The Points on L1 satisfy the equations of the planes P1 and P2. The Points on L2 satisfy the equations of the planes P2 and P3. The Points on L3 satisfy the equations of the planes P1 and P3. Let v1 be a vector along L1 which lies on both planes P1 and P2. Let v2 be a vector parallel to v1 and along L3 which lies on both planes P1 and P3. I think we could form vector v2 since vector v2 lies on P1 and vector v1 also lies on P1. Therefore the normals n1, n2 , and n3 to the planes P1, P2, and P3 are coplanar.
c) I am not sure how to transform my reasoning above algebraically.

 

[LCKurtz]

For (b), think about a vector parallel to each of the parallel lines i.e., a common direction vector $\vec D$ for the intersection lines. What is the relation of the three normals to the planes to $\vec D$? What does that tell you?
For (c), What direction would $\vec N_1 \times \vec N_2$ have? What if you dot that into $\vec N_3$? What do you know about the triple scalar product of 3 coplanar vectors?

 

[Vigorous]

b) The three normals are perpendicular to D. c) N1 × N2=rD where r is scalar. Since N3 is also normal to D.
rD.N3=0 (parallelepiped of volume 0) The three normals all lie on one plane.