Angle between a Line and a Plane

In this lesson on 2-D geometry, we define a straight line and a plane and how the angle between a line and a plane is calculated. Calculation methods in Cartesian form and vector form are shown and a solved example, in the end, is used to make the understanding easy for you.

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Two-Dimensional Geometry

In geometry, we use the term, “line” or “straight line” interchangeably. A straight line is a two-dimensional locus of an infinite number of points extending out in either direction. Hence, the length of a line is infinite and it has no height or width, making it a part of 2-D geometry.

We now turn to the plane. A plane is also a two-dimensional figure. When an infinite number of points extend infinitely in either direction to form a flat surface, it is called a plane. Thus, being a 2-D figure, a plane does not have any thickness but only length and width.

Browse more Topics Under Three Dimensional Geometry

• Angle Between a Line and a Plane
• Angle Between Two Lines
• Coplanarity of Two Lines
• Angle Between Two Planes
• Direction Cosines and Direction Ratios of a Line
• Distance Between Parallel Lines
• The Distance Between Two Skew Lines
• Distance of a Point from a Plane
• Equation of a Plane in Normal Form
• Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
• The Equation of Line for Space
• Equation of Plane Passing Through Three Non Collinear Points
• Intercept Form of the Equation of a Plane
• Plane Passing Through the Intersection of Two Given Planes

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The Angle between a Line and a Plane

We now move on to defining how to calculate the angle between a line and a plane. Note that when we refer to the plane and the line, in this case, we are actually referring to the angle between the normal to the plane and the straight line. This will be clear to you when you take a look at the following figure:

Source: GeoGebra

We now look at two ways of calculating the angle between a plane and a straight line.

Cartesian Form

We define the straight line equation in Cartesian form as follows:

(x – x₁)/ a = (y – y₁)/ b = (z – z₁)/ c

where, (x₁, y₁, z₁) represents the coordinates of any point on the straight line. The equation of a plane in Cartesian form is:

a₂x + b₂y + c₂z + d₂ = 0

where, (x₂, y₂, z₂) represents the coordinates of any point on the plane. Now, the angle between the line and the plane is given by:

Sin ɵ = (a₁a₂ + b₁b₂ + c₁c₂)/  a₁² + b₁² + c₁² ). ( a₂² + b₂² + c₂²)

Vector Form

Contrarily, the angle between a plane in vector form, given by r = a λ +b and a line, given in vector form as r * . n = d is given by:

Sin ɵ = n * b / |n| |b|

Solved Example for You

Question: Find the angle between the straight line (x + 1) / 2 = y/ 3 = (z – 3)/ 6 and the plane 10x + 2y – 11z = 3.

Answer: We can calculate the angle using the Cartesian form as under:
Sin ɵ = | 10 x 2 + 2 x 3 + (-11) x 6 | /  10² + 2² + (-11)² ). ( 2² + 3² + 6²)
Sin ɵ = | 20 + 6 – 66 | / (100 + 4 + 121). (4 + 9 + 36)
Sin ɵ = | – 40 | / (15 x 7)
Sin ɵ = 8/ 21
ɵ = Sin ^-1 (8/ 21) is the required angle between the plane and the line.

Question: Read the following statement carefully and identify the true statement

(a) Two lines parallel to a third line are parallel
(b) Two lines perpendicular to a third line are parallel
(c) Two lines parallel to a plane are parallel
(d) Two lines perpendicular to a plane are parallel
(e) Two lines either intersect or are parallel

1. a and b
2. a and d
3. d and e
4. a

Answer: Two lines parallel to a third line are parallel and two lines perpendicular to a plane are parallel. Hence the correct option is A and D.

Question: What is the perpendicular line?

Answer: In the geometry, the basic property of being perpendicular is the connection between 2 lines which meet at an angle of 90 degrees. The property outspreads to other related geometric entities. A line is perpendicular to another line when the 2 lines are intersecting at an angle of 90 degrees.

Question: What are the different kinds of lines?

Answer: Some different types of lines are a straight line, curved line, vertical line, horizontal line, etc. The different types of lines help the kids to understand the difference between these straight lines and the curved or slanted lines. The kids also get to know how the lines are drawn in various directions in geometry.

Question: What are the slanting lines?

Answer:  The straight line is vertical if it goes straight up and down, or top to bottom, without going diagonally or across at all. Lines are slanting if they are not going straight. These slanting lines look like a slope and go up and down both, and across too.

Question: How do we define a straight line?

Answer: A straight line is a set of all the points between and spreading beyond 2 points. In many geometries, a line is a basic object that does not have formal properties beyond its length. A straight line is always in a single dimension.