In geometry, it is important to know about various kinds of surfaces. A line is a one dimensional surface and a space is a three dimensional surface. However, a plane is a two dimensional surface with zero thickness. Planes have certain special properties which include:
Any two distinct planes are either parallel or intersect at a line.
A line may either lie within the plane or intersect the plane at a single point or parallel to the plane.
Two lines are parallel to each other if they are perpendicular to the same plane.
Two planes are also parallel to each other if they are perpendicular to the same line.
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The angle of separation of two intersecting planes is calculated as the angle of separation of normals to both the planes.
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Let us consider two planes intersecting at an angle θ as shown in the above figure. Let n1 and n2 be the normal vectors drawn to the planes. The equation for both the planes is thus given as:
\[\overrightarrow{r}\].\[\overrightarrow{n_{1}}\] = \[d_{1}\]
\[\overrightarrow{r}\].\[\overrightarrow{n_{2}}\] = \[d_{2}\]
Cosine of angle between two intersecting planes is given as the cosine of the angle between their normals:
Cosθ = \[\frac{n1.n2}{|n1||n2|}\] |
This is the angle between two planes formula when normal vectors are given.
Consider the angle between two planes example in which two planes intersecting at an angle θ. The equations of these two planes are given in the cartesian coordinate system as A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0. In these two equations, A1, B1 and C1 are the direction ratios of normal to the plane described by the equation A1x + B1y + C1z + D1 = 0 and A2, B2 and C2 are the direction ratios of normal to the plane defined by the equation A2x + B2y + C2z + D2 = 0. The cosine of the angle between the two planes is given as:
Cosθ = \[\frac{A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt{A_{1}^{2}+B_{1}^{2}+C_{1}^{2}}.\sqrt{A_{2}^{2}+B_{2}^{2}+C_{2}^{2}}}\] |
1. How to calculate angle between two planes described by the equations 2x + 4y - 4z - 6 = 0 and 4x + 3y + 9 = 0?
Solution:
In these equations of the plane,
A1 = 2, B1 = 4, C1 = - 4, D1 = - 6
A2 = 4, B2 = 3, C2 = 0, D2 = 9
Cosθ = \[\frac{A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt{A_{1}^{2}+B_{1}^{2}+C_{1}^{2}}.\sqrt{A_{2}^{2}+B_{2}^{2}+C_{2}^{2}}}\]
Cosθ = \[\frac{|2X4+4X3+(-4)X0|}{\sqrt{2^{2}+4^{2}+(-4)^{2}}.\sqrt{4^{2}+3^{2}+0^{2}}}\]
Cosθ = \[\frac{|8+12+0|}{\sqrt{4+16+16}.\sqrt{16+9+0}}\]
Cosθ = \[\frac{20}{\sqrt{36}.\sqrt{25}}\] = \[\frac{20}{6X5}\] = \[\frac{2}{3}\]
θ = \[Cos^{-1}(\frac{2}{3})\]
2. How to calculate angle between two planes when the direction vectors of normals of the planes are given as n1 = 2i + 4j - 2k and n2 = 6i - 8j - 2k.
Solution:
The coordinates of normal vector n1 is (2, 4, -2)
The coordinates of normal vector n2 is (6, -8, -2)
Cosθ = \[\frac{n1.n2}{|n1||n2|}\]
Cosθ = \[\frac{(2,4,-2)(6,-8,-2)}{|n1||n2|}\]
\[\frac{2\sqrt{39}}{\sqrt{4+16+4}.\sqrt{36+64+4}}\] = \[\frac{2\sqrt{39}}{39}\] = \[\frac{2}{\sqrt{39}}\]
θ = \[Cos^{-1}\frac{2}{\sqrt{39}}\]
Point is a dimensionless geometric shape.
A plane is formed by a stack of lines arranged side by side.
1. How to Calculate Angle Between Two Planes?
A plane is a two-dimensional geometrical surface. It is formed by stacking the straight lines one next to the other densely. When two planes intersect, the angle of separation of the planes is equal to the angle between the normals drawn to the planes. When two planes are described in cartesian coordinate system as A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0, the cosine of angle of separation between the two planes is given as:
Cosθ = |A₁A₂ + B₁B₂ + C₁C₂| / [√(A₁²+B₁²+C₁²).√(A₂²+B₂²+C₂²)].
2. How are Geometric Surfaces Categorized Based on Their Dimensions?
Geometric surfaces may have one, two or three dimensions.
A point is a dimensionless geometric surface.
A one-dimensional geometric surface is called a line. The line has only one dimension which is called length.
Plane is a two-dimensional geometric surface. The two dimensions of a plane are length and width.
A three-dimensional geometric surface is called space. The three dimensions of measuring a space are length, width, and height.
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