How to Draw 3 Intersecting Planes

The electric current inquiry tells us that there are 4 dimensions. These four dimensions are, x-airplane, y-plane, z-plane, and time. Since we are working on a coordinate system in maths, we will exist neglecting the time dimension for at present. These planes can intersect at any time at any place. There is no definite saying that whether they all will intersect or some of them will or possibly none of them might intersect each other. The question is how to place whether planes are intersecting with each other? Let's figure it out!

Intersection of Planes

The best and possible way to learn virtually their intersection is using the rank method. Below is a minor matrix of three 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 three planes, form a organization 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 between the three planes presents tin exist 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, as well equally the augmented matrix, volition be equal to 3.

r=3, r'=iii

2.1 Each Plane Cuts the Other Two in a Line.

So y'all have learned nigh a unmarried point, what if information technology was a line? And at that place might be a chance of two 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 iii.

r = ii, r' = 3

2.2 Two Parallel Planes and the Other Cuts Each in a Line

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

r = ii, r' = 3

2 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.ane Three Planes Intersecting in a Line

There is a possibility that all three planes will intersect each other simply not at a certain indicate but on the line. This can happen and the best way for its identification is that if the rank of the coefficient matrix, too as the augmented matrix, is equal to ii.

r = ii, r' = 2

3.ii Two Coincident Planes and the Other Intersecting Them in a Line

If two planes are coincident and the third aeroplane is intersecting in a mode that it creates a line then their rank of the coefficient matrix, likewise as the augmented matrix, will also exist equal to ii merely 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 = two, r' = 2

4.ane Three Parallel Planes

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

r = 1, r' = 2

four.2 Two Coincident Planes and the Other Parallel

If two planes are coincided and the 3rd ane is parallel and so the rank will exist the same, the rank of the coefficient matrix will exist one while the rank of the augmented matrix will be two. However, the point to detect is that the two rows of the augmented matrix will exist proportional, which is the indication that y'all are working with two coincident planes while the other plane is parallel.

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

r = one, r' = 2

5. Three Coincident Planes

Last but not least, are all three planes ancillary? Not an consequence! Both ranks, rank of the coefficient matrix also as rank of the augmented matrix, will be equal to one.

r = i, r' = 1

Examples

State the relationship between the three planes.

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

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

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

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

Each plane cuts the other ii in a line and they form a prismatic surface.

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

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

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

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

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 2nd and tertiary planes are ancillary and the first is cuting them, therefore the three planes intersect in a line.

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

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

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

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

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

The first and second are coincident and the third is parallel to them.

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Source: https://www.superprof.co.uk/resources/academic/maths/geometry/plane/intersection-of-three-planes.html

How to Draw 3 Intersecting Planes

Intersection of Planes

1. Intersecting at a Point

2.1 Each Plane Cuts the Other Two in a Line.

2.2 Two Parallel Planes and the Other Cuts Each in a Line

3.ane Three Planes Intersecting in a Line

3.ii Two Coincident Planes and the Other Intersecting Them in a Line

4.ane Three Parallel Planes

four.2 Two Coincident Planes and the Other Parallel

5. Three Coincident Planes

Examples

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