How to Draw 3 Planes That Intersect in One Line

mai 09, 2022 Enregistrer un commentaire

The current research tells us that at that place are 4 dimensions. These iv dimensions are, ten-plane, y-airplane, z-aeroplane, and time. Since we are working on a coordinate system in maths, we volition be neglecting the time dimension for now. These planes tin can intersect at any time at any place. In that location is no definite saying that whether they all will intersect or some of them volition or perhaps none of them might intersect each other. The question is how to identify whether planes are intersecting with each other? Let'southward figure it out!

Intersection of Planes

The best and possible way to learn about their intersection is using the rank method. Below is a small matrix of 3 planes.

$\left\{\begin{matrix} { A }_{ 1 } x + { B }_{ 1 } y + { C }_{ 1 } z + { D }_{ 1 } = 0 \\ { A }_{ 2 } x + { B }_{ 2 } y + { C }_{ 2 } z + { D }_{ 2 } = 0 \\ { A }_{ 3 } x + { B }_{ 3 } y + { C }_{ 3 } z + { D }_{ 3 } = 0 \end{matrix}\right$

To study the intersection of 3 planes, course a organisation with the equations of the planes and calculate the ranks.

r = rank of the coefficient matrix.

r'= rank of the augmented matrix.

There is a lot of possibilities for plane intersections. That is why we listed all kinds of possibilities and their identifications. The relationship betwixt the three planes presents can be described as follows:

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Let's go

1. Intersecting at a Point

When all three planes intersect at a single point, their rank of the coefficient matrix, too equally the augmented matrix, volition be equal to three.

r=3, r'=3

ii.one Each Aeroplane Cuts the Other Two in a Line.

So you have learned near a single signal, what if information technology was a line? And at that place might exist a risk of 2 lines intersection too. This type of intersection volition create a prismatic surface. The rank of the coefficient matrix will be two, however, the rank of the augmented matrix will be equal to three.

r = 2, r' = iii

2.two Two Parallel Planes and the Other Cuts Each in a Line

Same line scenario but a single plane cuts both parallels planes making a line intersection. The rank of the coefficient matrix will be two while the rank of the augmented matrix will exist three.

r = 2, r' = 3

Two rows of the coefficient matrix are proportional. This is an identification of two parallel planes and the other cuts each in a line.

$\frac { A }{ { A }^{ t } } = \frac { B }{ { B }^{ t } } = \frac { C }{ { C }^{ t } } \neq \frac { D }{ { D }^{ t } }$

3.1 Iii Planes Intersecting in a Line

In that location is a possibility that all three planes will intersect each other but not at a certain point but on the line. This can happen and the best manner for its identification is that if the rank of the coefficient matrix, likewise as the augmented matrix, is equal to two.

r = 2, r' = 2

3.2 2 Coincident Planes and the Other Intersecting Them in a Line

If ii planes are coincident and the third plane is intersecting in a mode that it creates a line then their rank of the coefficient matrix, every bit well equally the augmented matrix, will also be equal to two just with a twist. The two rows of the augmented matrix volition be proportional.

$\frac { A }{ { A }^{ t } } = \frac { B }{ { B }^{ t } } = \frac { C }{ { C }^{ t } } = \frac { D }{ { D }^{ t } }$

r = 2, r' = ii

4.1 Three Parallel Planes

What if none of those planes intersects at any point but they are parallel? Then their rank of the coefficient matrix will be one, however, the rank of the augmented matrix will be two.

r = i, r' = 2

4.2 Two Coincident Planes and the Other Parallel

If two planes are coincided and the third one is parallel and then the rank will exist the aforementioned, the rank of the coefficient matrix volition be one while the rank of the augmented matrix will be 2. Withal, the bespeak to observe is that the two rows of the augmented matrix will be proportional, which is the indication that you are working with two ancillary planes while the other plane is parallel.

$\frac { A }{ { A }^{ t } } = \frac { B }{ { B }^{ t } } = \frac { C }{ { C }^{ t } } = \frac { D }{ { D }^{ t } }$

r = 1, r' = 2

5. Three Coincident Planes

Terminal just not least, are all three planes coincident? Not an event! Both ranks, rank of the coefficient matrix as well as rank of the augmented matrix, will be equal to ane.

r = one, r' = 1

Examples

Land the human relationship between the three planes.

ane. $\begin{matrix} { \pi }_{ 1 } \equiv x + y - z + 3 = 0 \\ { \pi }_{ 2 } \equiv -4x + y + 4z - 7 = 0\\ { \pi }_{ 3 } \equiv -2x + 3y + 2z - 2 =0 \end{matrix}$

$\begin{matrix} x + y - z = -3 \\ -4x + y + 4z = 7 \\ -2x + 3y + 2z = 2 \end{matrix}$

$M = \begin{pmatrix} 1 & 1 & -1 \\ -4 & 1 & 4 \\ -2 & 3 & 2 \end{pmatrix} \quad \begin{vmatrix} 1 & 1 & -1 \\ -4 & 1 & 4 \\ -2 & 3 & 2 \end{vmatrix} = 0 \qquad r = 2$

${ M }^{ t } = \begin{pmatrix} 1 & 1 & -1 & -3 \\ -4 & 1 & 4 & 7 \\ -2 & 3 & 2 & 2 \end{pmatrix} \quad \begin{vmatrix} 1 & 1 & -3 \\ -4 & 1 & 7 \\ -2 & 3 & 2 \end{vmatrix} \neq 0 \qquad { r }^{ t } = 3$

Each airplane cuts the other 2 in a line and they grade a prismatic surface.

two. $\begin{matrix} { \pi }_{ 1 } \equiv 2x - 3y + 4z - 1 = 0 \\ { \pi }_{ 2 } \equiv x - y - z + 1 = 0\\ { \pi }_{ 3 } \equiv -x + 2y - z + 2 =0 \end{matrix}$

$\begin{matrix} 2x - 3y + 4z = 1 \\ x - y - z = -1 \\ -x + 2y - z = -2 \end{matrix}$

$M = \begin{pmatrix} 2 & -3 & 4 \\ 1 & -1 & -1 \\ -1 & 2 & -1 \end{pmatrix} \quad \begin{vmatrix} 2 & -3 & 4 \\ 1 & -1 & -1 \\ -1 & 2 & -1 \end{vmatrix} \neq 0 \qquad r = 3$

${ M }^{ t } = \begin{pmatrix} 2 & -3 & 4 & 1 \\ 1 & -1 & -1 & -1 \\ -1 & 2 & -1 & -2 \end{pmatrix} { r }^{ t } = 3$

Each plan intersects at a point.

3. $\begin{matrix} { \pi }_{ 1 } \equiv 2x + 3y + z - 1 = 0 \\ { \pi }_{ 2 } \equiv x - y + z + 2 = 0\\ { \pi }_{ 3 } \equiv 2x - 2y + 2z + 4 =0 \end{matrix}$

$\begin{matrix} 2x - 3y + z = 1 \\ x - y - z = -2 \\ 2x - 2y + 2z = -4 \end{matrix}$

$M = \begin{pmatrix} 2 & 3 & 1 \\ 1 & -1 & 1 \\ 2 & -2 & 2 \end{pmatrix} \quad \begin{vmatrix} 2 & 3 & 1 \\ 1 & -1 & 1 \\ 2 & -2 & 2 \end{vmatrix} = 0 \qquad r = 2$

${ M }^{ t } = \begin{pmatrix} 2 & -3 & 1 & 1 \\ 1 & -1 & 1 & -2 \\ 2 & -2 & 2 & -4 \end{pmatrix} \qquad \begin{vmatrix}2 & -3 & 1 \\ 1 & -1 & -2 \\ 2 & -2 & -4 \end{vmatrix} = 0 \qquad { r }^{ t } = 2$

$\frac { 1 }{ 2 } = \frac { -1 }{ -2 } = \frac { 1 }{ 2 } = \frac { -2 }{ -4 }$

The second and third planes are coincident and the first is cuting them, therefore the three planes intersect in a line.

4. $\begin{matrix} { \pi }_{ 1 } \equiv 2x - y + 2z + 1 = 0 \\ { \pi }_{ 2 } \equiv -4x + 2y - 4z - 2 = 0\\ { \pi }_{ 3 } \equiv 6x - 3y + 6z + 1 =0 \end{matrix}$

$\begin{matrix} 2x - y + 2z = -1 \\ -4x + 2y - 4z = 2 \\ 6x - 3y + 6z = -1 \end{matrix}$

$M = \begin{pmatrix} 2 & -1 & 2 \\ -4 & 2 & -4 \\ 6 & -3 & 6 \end{pmatrix} \quad \begin{vmatrix} 2 & -1 & 2 \\ -4 & 2 & -4 \\ 6 & -3 & 6 \end{vmatrix} = 0 \qquad \begin{vmatrix} 2 & -1 \\ -4 & 2 \end{vmatrix} \qquad r = 1$

${ M }^{ t } = \begin{pmatrix} 2 & -1 & 2 & -1 \\ -4 & 2 & -4 & 2 \\ 6 & -3 & 6 & -1 \end{pmatrix} \qquad \begin{vmatrix} 2 & -1 & -1 \\ -4 & 2 & 2 \\ 6 & -3 & -1 \end{vmatrix} = 0 \qquad \begin{vmatrix} 2 & 2 \\ -3 & -1 \end{vmatrix} \neq 0 \qquad { r }^{ t } = 2$