# Relationship between lines and planes

### Line-Plane Intersecting

So both vectors are perpendicular to the given plane. calculate the cross product between the normal vector and the parallel vector from line. 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. Shapes 5 Line & Plane Relationships Name_______________________ Worksheet A 1. Name all segments parallel to GE. 2. Name all segments parallel to.

Here, we describe that concept mathematically. Remember, the dot product of orthogonal vectors is zero.

This fact generates the vector equation of a plane: As described earlier in this section, any three points that do not all lie on the same line determine a plane. Given three such points, we can find an equation for the plane containing these points.

Solution To write an equation for a plane, we must find a normal vector for the plane. We start by identifying two vectors in the plane: We want to find the shortest distance from point P to the plane. Just as we find the two-dimensional distance between a point and a line by calculating the length of a line segment perpendicular to the line, we find the three-dimensional distance between a point and a plane by calculating the length of a line segment perpendicular to the plane.

When we describe the relationship between two planes in space, we have only two possibilities: When two planes are parallel, their normal vectors are parallel.

The intersection of two nonparallel planes is always a line. We can use the equations of the two planes to find parametric equations for the line of intersection. Solution Note that the two planes have nonparallel normals, so the planes intersect. Further, the origin satisfies each equation, so we know the line of intersection passes through the origin.

For example, builders constructing a house need to know the angle where different sections of the roof meet to know whether the roof will look good and drain properly. We can use normal vectors to calculate the angle between the two planes. We can do this because the angle between the normal vectors is the same as the angle between the planes.

The angle between two planes has the same measure as the angle between the normal vectors for the planes. Finding the Angle between Two Planes Determine whether each pair of planes is parallel, orthogonal, or neither.

If the planes are intersecting, but not orthogonal, find the measure of the angle between them. Give the answer in radians and round to two decimal places.

The normal vectors are parallel, so the planes are parallel. Hint Use the coefficients of the variables in each equation to find a normal vector for each plane. To find this distance, we simply select a point in one of the planes.

## Line–plane intersection

The distance from this point to the other plane is the distance between the planes. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product.

Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes?

Again, this can be done directly from the symmetric equations. Let's say I had a point, B, right over here. Well, notice the way I drew this, point A and B, they would define a line. For example, they would define this line right over here.

So they would define, they could define, this line right over here.

## Parallel and Perpendicular Lines and Planes

But both of these points and in fact, this entire line, exists on both of these planes that I just drew. And I could keep rotating these planes. I could have a plane that looks like this. I could have a plane that looks like this, that both of these points actually sit on. I'm essentially just rotating around this line that is defined by both of these points.

So two points does not seem to be sufficient. So there's no way that I could put-- Well, let's be careful here. So I could put a third point right over here, point C. And C sits on that line, and C sits on all of these planes. So it doesn't seem like just a random third point is sufficient to define, to pick out any one of these planes. But what if we make the constraint that the three points are not all on the same line.

Obviously, two points will always define a line. But what if the three points are not collinear.

**Geometry Help from francinebavay.info - Point Line Plane**

So instead of picking C as a point, what if we pick-- Is there any way to pick a point, D, that is not on this line, that is on more than one of these planes?

If I say, well, let's see, the point D-- Let's say point D is right over here. So it sits on this plane right over here, one of the first ones that I drew. So point D sits on that plane. Between point D, A, and B, there's only one plane that all three of those points sit on.

So a plane is defined by three non-colinear points.

So D, A, and B, you see, do not sit on the same line. A and B can sit on the same line.

### An introduction to geometry (Geometry, Points, Lines, Planes and Angles) – Mathplanet

D and A can sit on the same line. D and B can sit on the same line.

But A, B, and D does not sit on-- They are non-colinear. So for example, right over here in this diagram, we have a plane.