Intersection of a line and a plane

What Does The Term “Intersecting Lines” Mean?

When two or more lines intersect in a plane, “intersecting lines” are used. It is called the junction point because the intersecting lines have a common point present in all of them.

When two lines P and Q connect, a point called the intersection of a line and a plane is formed. When two lines share a common meeting point, they intersect. The common point in all the lines is called the point of intersection of all the lines. Various methods can prove this point of interaction. 

Let us now learn about the intersection of a line and a plane, features of intersecting lines, and more. 

The Intersection of a line and a plane

A line is a group of infinite points joining together endlessly in opposing directions. It has just one dimension, which is its length. Collinear points are those that are parallel to one another.

A point is an undetermined location on a plane that lacks dimensions, i.e., it has no width, length, or depth.

Between two planes in a three-dimensional space, the following connections may be created:

• They may be next to one another.
• They might be identical or dissimilar.
• They may come into contact through a line of intersection of the planes.

How To Find Where a Line Intersects a Plane?

Q) find the intersection point formed by the line and plane with the following equations in     parametric and scalar forms, respectively.

2x + y – 2z = 4

x = 1+ t ,y= 4 + 2t , z=t

Ans:- 

The equation of the line is in its parametric form and the equation of the plane is in scalar form. This means that we can use the parametric form of the line’s equation to rewrite the      scalar equation of the plane.

2x + y – 2z = 4  

 2(1+ t) + (4 + 2t) – 2(t) = 4

Simplify the resulting expression then solve for the parameter, t.

2+ 2t + 4 + 2t – 2t = 4

2t +6 = 4

2t=-2

 t= -1

Use the parametric equations of the line and t=−1 to find the components of the point.

x = 1+ (-1)

= 0

y= 4 + 2(-1)

=2

 z=-1

(x, y, z) = (0, 2, -1)

This means that the line and the plane will intersect at the point, (0,2,−1)

Intersecting lines have certain features:

• Two or more intersecting lines converge at a single location.
• The crossing lines may be perpendicular to one another. However, the magnitude of the resulting angle is always more than 0 and less than 1800.
• When two intersecting lines meet, they create a pair of vertical angles. Vertical angles are diametrically opposite angles and share a common vertex (the point of intersection).

Fascinating Fact

When two or more lines intersect at several locations, curved lines are generated rather than straight lines.

In the intersection of a line and a plane, the perpendicular lines are defined as two lines that intersect at an exact angle of 90 degrees (creating a perpendicular line). As a consequence, perpendicular lines are a subset of different intersecting lines. 

Now that we know that perpendicular lines are a subset of different intersecting lines let us look at a couple of examples of intersecting lines that we can observe daily and are around us all the time. 

Scissors are a perfect example of intersecting lines. Similarly, crossroads or roads that intersect are also good examples of intersecting lines. You can spot intersecting lines at various other places. Just keep an open mind and be always willing to learn and grow.

Uses

A surface may be represented as a collection of linked planes using the computer graphics technique of ray tracing. A surface picture is formed by crossing a light beam with each surface’s planes. Deep values are often evaluated using the so-called triangulation approach, which locates the intersection of the light plane, and a ray reflected toward the camera in computer vision-based 3D reconstruction, one of the several subfields of computer vision.

The approach may be extended to include intersections with other planar figures, such as the intersection of a polyhedron and a line. Now that we know all about intersecting lines, let us jump onto the next part, frequently asked questions. These questions will help you brush up on your concepts and revise what you’ve learned. 

Conclusion 

We have learnt that the term “intersecting lines” is used when two or more lines intersect in a plane and combine to create a single line. It is referred to as the point of the junction because the crossing lines have a common point that is present on all of them, therefore earning it the name.

When the lines P and Q come together, a point known as the intersection of a line and a plane is produced in the middle of them. A line is a collection of points that may be connected indefinitely in opposite directions by joining them together. It only has one dimension, and that is the length of the thing in question. Points that are parallel to one another are referred to as collinear points.

An unknown place on a plane with no dimensions, i.e. no width, no length, and no depth is referred to as a point.