Intersection of a Straight Line and a Plane

The intersection of a straight line and plane focuses on the relation between line and plane equations in a three-dimensional coordinate system. The intersection of a straight line and a plane aids in understanding equation intersections in R2 to R3.

In a three dimensional space, an intersection can be:

• An empty set 
• A point 
• A line 

A point is the meeting of a line and a plane.

P (xo, yo, zo) satisfies the equation of the line and the plane in R3. 

What is a Line and a Plane? 

In mathematics, a line is a straight one-dimensional figure. It is often defined as the shortest distance between two points, and it has no curves. A plane, on the other hand, is a two-dimensional figure. It is defined as a flat surface that extends indefinitely and has two linear independent vectors. 

Equation of a Straight Line and Intercept Form 

The equation of a straight line is:
ax+by+c= 0

Here: 

a, b and c are constants
x and y are variables

Equation of intercept form with x intercept as a and y intercept as b is: 

xa + yb =1 

What are the Possibilities when a Line and a Plane Intersect ?

When the line lies on a plane, there will be three possible intersections: 

• If the line is in a plane, there are an infinite number of intersections between the line and the plane. 

• If the line is parallel to the plane, the line and plane will not intersect. 

• If the line intersects the plane once, the line and the plane intersect.

Lines that are Parallel and Plane 

A plane can be expressed as the set of points p (in a vector notation). If the line is parallel to the plane, the normal vector and plane are also at a right angle to the direction vector v of the line. 

Product of n.v = 0 

This confirms that the two vectors are perpendicular to each other and that there is no interaction.

Intersecting Lines and Planes 

The interaction of a line and a plane will have a common meeting point. 

The parametric equation of a line is 

 x=( x0 + at , y= y0+ bt, z= z0 +ct )

The scalar equation for a plane: 

ax + by + cz + d = 0

When a straight line and plane cross at one point:

a ( x0 +at ) + b ( y0 + bt ) + c ( z0 + ct) + d = 0

In the equation above, the parameter “t” is found.

Components to Find the Line and Plane Intersection

These are the steps to find the line and plane intersection:

• First, write down the calculation of the line in the variable form: 

x=( x0 + at , y= y0+ bt, z= z0 +ct )

• Write the calculation of the plane in its scalar form:

Ax + By + Cz + D = 0

• Rewrite the scalar equation of the plane with x, y, and z to the variable calculation.

   

• Substitute t in the variable calculation. This way x, y, and z components are established.

   

• The single-variable equation thus helps us solve t.

Examples of Straight Line and Point Questions 

A line intersects with a plane in the following calculation, let us find the meeting point.

2x + y -2 z = 4

x= 1+ t

y= 1+ 2t

z= t 

We now rewrite the scalar equation of the plane, using the parametric forms:

2x +y – 2z = 4 

2 ( 1+t ) + ( 4+ 2t) – 2 (t ) = 4

Let us now solve to get the value of t:

2+ 2t + 4 + 2t- 2t = 4

2t + 6 = 4

2t =  -2

t= -1

Using t= -1 , and the parametric equation:

x= 1 + ( -1 )

= 0

y= 4 + 2 ( -1 )

=2. 

Hence x, y, z =  (0, 2, -1). 

The intersection of the straight line and plane is 0, 2, -1. 

Conclusion

By solving straight line and a plane questions, we have ample information to understand all about this section of analytic geometry.  Apart from mathematics, the intersection of a straight line and a plane finds its uses in the ray tracing method in computer graphics.

It is also used in vision-based 3D reconstruction, like video game programming, by making use of the algorithm of the intersection of a straight line with a polyhedron.