When an infinite amount of points extends in only two directions, we call it a line. A line has only one dimension and that is the length. However, a plane is something close to a line. A plane is a two-dimensional surface and like a line, it extends up to infinity. Usually, we talk about the line-line intersection. It means that two or more than two lines meet at a point or points, we call those point/points intersection point/points. The same concept is of a line-plane intersection. It means that when a line and plane comes in contact with each other. That point will be known as a line-plane intersection.

Line-plane and line-line are not the only intersections in geometry, you will also find line-point intersection as well. There is one problem, how can you identify whether the intersection is a line-line, line-point, or no intersection without the help of the graph? Well, there is one method through which you can find the type of intersection. You can find it with the help of the ranks.

Line Defined by Two Planes

The lines are r \equiv \begin{cases} { A }_{ 1 }x  + { B }_{ 1 }y + { C }_{ 1 }z + { D }_{ 1 } = 0\\ { A }_{ 2 }x  + { B }_{ 2 }y + { C }_{ 2 }z + { D }_{ 2 } = 0 \end{cases} and the plane is \pi \equiv { A }_{ 3 }x  + { B }_{ 3 }y + { C }_{ 3 }z + { D }_{ 3 } = 0

Form a system with equations in order to calculate the ranks.

\begin{cases} { A }_{ 1 }x  + { B }_{ 1 }y + { C }_{ 1 }z + { D }_{ 1 } = 0 \\ { A }_{ 2 }x  + { B }_{ 2 }y + { C }_{ 2 }z + { D }_{ 2 } = 0 \\ { A }_{ 3 }x  + { B }_{ 3 }y + { C }_{ 3 }z + { D }_{ 3 } = 0 \end{cases}

The next step is to calculate the coefficient matrix and augmented matrix and compare the values with values of different types of intersection. Here r means the rank of the coefficient matrix and r' means rank of the augmented matrix.

The relationship between the line and the plane can be described as follows:

Point Intersection

The basic condition for the intersecting line is that both lines should cut each other on a single point.

The identification of intersecting lines is that the rank of the matrix will always be equal to the rank of the augmented matrix and they both should be equal to 3 i.e. r = r' = 3. For example, in the below plane diagram, the rank of the augmented matrix, as well as the rank of the coefficient matrix, is 3 which means r = r' = 3

No Intersection

When two or more than two lines don't meet each other (or cut each other) that means you have a case of no intersection. The rank of the coefficient matrix will always be equal to 2, however, the rank of the augmented matrix will be equal to 3. Don't get this confused with skew lines. If r =3 \qquad r' = 4, that indicates the lines are skew lines, on the other hand, no intersection means r = 2 \quad r' = 3. Furthermore, skew line occurs in two planes, no intersection can occur in a single plane.

Line Intersection

Imagine a line and then draw another line which is overlapping the first line. In this case, at every point, there will be an intersecting point. The result will be that there will be an infinite number of intersection points, therefore, we will say that there will be an infinite number of intersecting points. This type of intersection is called line intersection and lines are called coincident lines.

At this point, you might be wondering what is the difference between intersecting lines and coincident lines, the only difference is that the intersecting line will always intersect at a single point, however, coincident lines intersect at many points that we make it equal to infinity. Furthermore, the coefficient rank of coincident lines will be r = 2 and the rank of the augmented matrix will be equal to r' = 2 i.e. r = r' = 2

Parallel Lines

Parallel lines mean both lines have a constant distance and they don't meet. Draw a line of x = 1 and draw another line of x = 4, these vertical lines will never meet each other but lines are identical to each other, except they are at a distance. This distance will remain constant. The rank of the parallel line will always be equal to 2, however, the rank of the augmented matrix will always be equal to 3 i.e. r = 2 \qquad r' = 3.

Examples

State the relationship between the line and the plane:

1.  r \equiv \frac { x + 1 }{ 2 } = \frac { y }{ 1 } = \frac { z }{ -1 } \qquad \pi \equiv x - 2y + 3z + 1 = 0

Form the system of equations.

\begin{cases} x - 2y = -1 \\ y + z = 0 \\ x - 2y + 3z = -1 \end{cases}

Find the rank of the coefficient matrix.

M = \begin{pmatrix} 1 & -2 & 0 \\ 0 & 1 & 1 \\ 1 & -2 & 3 \end{pmatrix}

\begin{vmatrix} 1 & -2 & 0 \\ 0 & 1 & 1 \\ 1 & -2 & 3 \end{vmatrix} \neq 0

r = 3

Determine the rank of the augmented matrix.

M' = \begin{pmatrix} 1 & -2 & 0 & -1 \\ 0 & 1 & 1 & 0 \\ 1 & -2 & 3 & -1 \end{pmatrix}

r' = 3

Compare the ranks.

r = 3 \qquad r' = 3

Point intersection.

 

2. r \equiv \frac { x - 1 }{ 5 } = \frac { y }{ 1 } = \frac { z + 2 }{ 1 } \qquad \pi \equiv -x + 3y + 2z + 5 = 0

\begin{cases} x - 5y = 1 \\ y - z = 2 \\ - x + 3y + 2z = -5 \end{cases}

M = \begin{pmatrix} 1 & -5 & 0 \\ 0 & 1 & -1 \\ -1 & 3 & 2 \end{pmatrix}

\begin{vmatrix} 1 & -5 & 0 \\ 0 & 1 & -1 \\ -1 & 3 & 2 \end{vmatrix} = 0

r = 2

Determine the rank of the augmented matrix.

M' = \begin{pmatrix} 1 & -5 & 0 & 1 \\ 0 & 1 & -1 & 2 \\ -1 & 3 & 2 & -5 \end{pmatrix}

\begin{vmatrix} 1 & -5 & 1 \\ 0 & 1 & 2 \\ -1 & 3 & -5 \end{vmatrix} \neq 0

r' = 3

Compare the ranks.

r = 2 \qquad r' = 3

The line and plane are parallel.

 

The Line is Defined by a Point and a Vector

The line is defined by Point A, the vector, \vec { u } and the plane by the normal vector, \vec { n } .

Intersection\vec { u } . \vec { n }A
Line intersection=0\in \pi
No intersection= 0\notin \pi
Point intersection\neq 0
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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.