In geometry, a line and an angle are used with a different meaning. A line in geometry is a set of closely arranged points that extend lengthwise in both the directions. When a set of lines are arranged adjacent to each other, a plane is obtained. A plane is a geometric surface that has two dimensions. The line has only one dimension which is measured in terms of length whereas a plane is a two-dimensional surface measured in terms of length and width. When a line is an incident on a plane, it forms an angle with the plane at the point of contact. This angle is said to be the angle between a line and a plane.

When a line is inclined to a plane at a certain angle and touches the plane at a particular point, the angle between the line and plane is equivalent to the complement of the angle between the line and the normal to the plane at the point of contact of line and plane.

Consider a line indicated in the above diagram in brown color. Let vector ‘n’ represent the normal drawn to the plane at the point of contact of line and plane. Let the angle between the line and the plane be ‘α’ and the angle between the line and the normal to the plane be ‘β’. The angle between a line and a plane as represented in the figure is equal to the complement of the angle between the line and the normal to the plane. i.e. the angle ‘α’ is equal to the complement of the angle ‘β’. In other words, the value of angle ‘α’ is equal to the value obtained when the value of angle ‘β’ is subtracted from 900.

Consider a line and a plane in which the line is inclined to the plane at a certain angle. A line is represented by a vector equation as where ‘a’ is the position vector of the initial point of the line and r is the position vector of the endpoint of the line. ‘b’ indicates the direction vector of the line drawn from the initial to the final point. Similarly, the vector equation of a plane is . Let the angle between the line and the normal is ‘θ’ and the angle between the line and the plane is ‘Φ’. The cosine of the angle between the line and the normal is given as:

\[\cos \theta = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|\]

The line drawn normal to the plane is always perpendicular to the plane. The sum of the angle between the line and the plane and the angle between the line and the normal is equal to the right angle. So, the angle between the line and a plane is equal to the complement of the angle between the line and the normal. From the formulas of trigonometric ratios of complementary angles, Cos θ = Sin (90 - θ) = Sin Φ. So, the sine of the angle between the line and a plane is given by the equation for Cos θ as:

\[\sin \Phi = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|\]

So, the angle between a line and a plane is given as:

\[\Phi = {\sin ^{ - 1}} = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|\]

Find the angle between line and plane where the line is represented by \[r = \frac{{x + 1}}{2} = \frac{{y + 1}}{1} = \frac{z}{2}\]and the equation of the plane is x + y - 1 = 0. (Hint: Use the angle between line and plane formula)

Solution:

Let us consider the angle between the line and the plane as Φ.

The given line can be written in vector form as

\[\hat r = \left( {i - j} \right) + \lambda \left( {2i + j + 2k} \right)\]

The equation of the normal to the plane in vector form is given as:

\[\hat r = i + j\]

So, b = 2i + 1j + 2k and n = i + j

The sine of the angle between the line and the plane is given as:

\[\sin \Phi = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|\]

Substituting the values of b and n, we get

\[\sin \Phi = \left| {\frac{{\left( {2i + 1j + 2k} \right)\left( {i + j - k} \right)}}{{\sqrt {{2^2} + {1^2} + {2^2}} .\sqrt {{1^2} + {1^2} + {0^2}} }}} \right|\]

\[Sin\phi\;=|\frac{2\;.\;1\;+\;1\;.\;1\;+\;2\;.\;0}{\sqrt{9}\;.\;\sqrt{2}}|\]

\[Sin\phi\;=|\frac{3}{3\;.\;\sqrt{2}}|\]

\[Sin\phi\;=|\frac{1}{\sqrt{2}}|\]

\[\phi\;=Sin^{-1}(\frac{1}{\sqrt{2}})\]

\[\phi\;=45^{0}\]

So, the angle between the line and the plane is 450.

When a line is parallel to a plane or on the plane, the line does not form any angle with the plane and the line is perpendicular to the normal drawn to the plane.

If a line is perpendicular to any two lines on the same plane, then that line is perpendicular to the plane also.

FAQ (Frequently Asked Questions)

1. How Do You Find the Angle Between a Line and a Plane?

A line is a single dimension geometric shape. Length is the only dimension that describes a line. A plane is a two-dimensional geometric figure that can be described in terms of length and width. When a line is an incident on the plane at a certain angle of inclination, this angle of inclination gives the angle between the line and a plane. To calculate the angle between a line and a plane a normal is drawn perpendicular to the plane at the point of contact with the line. The angle between the line and the plane is calculated as the complement of the angle between the line and the normal drawn to the plane. The angle between line and plane formula is:

2. Is Planar Geometry and Solid Geometry the Same?

No, planar geometry and solid geometry are entirely two different perceptions of an object. Planar geometry describes any object only in two dimensions in terms of its length and width. However, solid geometry describes the object in 3 dimensions. It gives the position of the object with respect to space in terms of length, width, and height. Planar geometry deals with flat objects such as triangles, circles, quadrilaterals, and other polygons. Solid geometry deals with the study of 3-dimensional shapes such as cubes, cuboids, spheres, prisms, pyramids, cones, cylinders, etc.