# 3 Different and not parallel planes

• Vigorous
In summary, the three given parallel lines L1, L2, and L3 have corresponding planes P1, P2, and P3 respectively. A vector v1 along L1 lies on both P1 and P2, while a vector v2 parallel to v1 and along L3 lies on both P1 and P3. This leads to the conclusion that the normals n1, n2, and n3 to the planes P1, P2, and P3 are coplanar. From here, considering a common direction vector for the intersection lines, it can be deduced that the three normals are perpendicular to this vector. Algebraically, this can be represented as N1 × N2 = rD where r
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.

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?

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.

## 1. What are 3 different and not parallel planes?

Three different and not parallel planes are three flat surfaces in three-dimensional space that do not intersect and are not parallel to each other. This means that they do not lie in the same plane and do not have any points in common.

## 2. How do you determine if three planes are not parallel?

To determine if three planes are not parallel, you can use the dot product or cross product of their normal vectors. If the dot product is zero, the planes are parallel. If the cross product is zero, the planes are either parallel or identical.

## 3. Can three planes be different but still parallel?

No, three planes cannot be different and parallel. If three planes are parallel, they must have the same slope and never intersect. If they are different, they must have different slopes and will eventually intersect.

## 4. What is the relationship between three different and not parallel planes?

The relationship between three different and not parallel planes is that they are skew lines, meaning they are not parallel and do not intersect, but they are not in the same plane. They also form three pairs of parallel planes, with each pair having one plane from each of the three original planes.

## 5. How can three different and not parallel planes be used in real life?

Three different and not parallel planes can be used in real life in various fields such as architecture, engineering, and mathematics. For example, in architecture, they can be used to create interesting and unique building designs. In engineering, they can be used to create stable structures. In mathematics, they can be used to solve problems involving three-dimensional geometry and vectors.

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