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Geometry is one of the most important sections of mathematics. Among them, 3D geometry is the study of the properties of a line and plane together making an angle with each other.
- An unlimited number of points form the two-dimensional locus of a straight line, which extends outward in both directions.
- A plane is a flat surface created when an unlimited number of points stretch endlessly in any direction.
- Line falls under the category of 2D geometry, for having no height or width.
- Plane also falls under the category of 2D geometry, for having no thickness.
- The equations of lines and planes under the angle between a line and a plane can be explained in the vector form and the Cartesian form.
- The direction cosines and direction ratios of a line joining two points can also be observed by these equations.
Also Read: Three-Dimensional Geometry
Key Terms: Three-dimensional plane, Vectors, Magnitude, Dimensions, One-dimensional geometry, Angle equation, Distance.
What Do a “Line” and a “Plane” Mean?
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A line is a one-dimensional figure, which has length but no width. A line is made of a set of points which is extended in opposite directions infinitely.
- Similarly, when an infinite number of points extend infinitely in either direction to form a flat surface, it is called a plane.
- A set of lines when arranged adjacent to each other a plane is obtained.
- A plane is a two-dimensional geometric figure that can be measured in terms of length and width.
Read Also: Addition of Vectors
What Does an Angle between a “Line” and a “Plane” Mean?
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The complement of the angle between the line which is adjacent to the plane and the normal of the plane is called the angle between a line and a plain.
Angle between a line and a plane
Vector equation of any life is given by -
Similarly, vector equation of a plane is given by –
Let, be the angle between the line and the normal of the plain. Then,
Now, let the line be inclined at an angle µ to the plane.
By the definition , µ is the complement of the angle .
And , sin 90°-θ =sin µ =cosθ .
Thus it can be written :
or, 
Read Also: Co Planar Vectors
Direction Cosines and Direction Ratios of a Line
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Directional angles: The angles made by a line with the positive directions of the X, Y, and Z axes are called directional angles.
Directional cosines: If α, β, and γ are the directional angles of a directed line L, then cosα,cosβ and cosγ are called the directional cosines of the line.
Directional cosines
Direction ratio of the line joining two points A(x1,y1,z1) and B(x2,y2,z2) is given by
(x2−x1,y2−y1,z2−z1)
If a vector is given by A=pi+qj+rk, then its direction ratios are given by (p,q,r)
Read Also: Coordinate Geometry
Relation between Direction Cosines and Direction Ratios
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If l, m,n are the direction cosines of a line, then a, b, c are its direction ratios such that :
and l2+m2+n2=1
- Cartesian forms of a straight line and a plane
\(\frac{(x-x_{1})}{a}=\frac{(y-y_{1})}{b}=\frac{(z-z_{1})}{c}\) - Cartesian form of a line.
a2x + b2y + c2z + d2 = 0 – Cartesian form of a plane.
Therefore, the angle between the line and the plane is given by :
Read More: Equation of a Plane in Three-Dimensional Space
Solved Examples
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Example 1: Find the angle between the line \(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z+2}{4}\) and the plane 2x+y-3z+4=0.
Solution: Here,
Equation of the line :
\(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z+2}{4}\)Equation of the plane is 2x+y-3z+4=0
We know, \(cos(\frac{π}{2}−θ)=\frac{6+2-12}{\sqrt{9+4+6}\sqrt{4+1+9}}\)
Let, the required angle be θ.
\(sinθ=\frac{-4}{\sqrt{29X14}}=\frac{-4}{\sqrt{406}}\) \(θ=sin−1(-\tfrac{4}{\sqrt{406}})\)Example 2: Find out the angle between the plane 10x + 2y - 11z = 3 and the straight line (x +1)/ 2 =y/ 3 = (z - 3)/ 6.
Solution: The angle can be calculated using the Cartesian form like,
Sinθ = | 10 x 2 + 2 x 3 + (-11) x 6 | / 102 +22 + (-11)2). ( 22 + 32 + 62)
Sinθ = | 20 + 6 - 66 | / ( 100 + 4 + 121). ( 4 + 9 + 36)
Sinθ = | -40 | / ( 15 x 7)
Sinθ = 8/ 21
θ = Sin-1 (8/21) is the angle between the plane and the line.
Things to Remember
- The two-dimensional locus of a straight line, which extends outward in both directions, is formed by an infinite number of points.
- When an infinite number of points extend eternally in any direction, a plane is formed.
- Since a line has neither height or breadth, it is said to be in two dimensions.
- A plane is a kind of 2D geometry since it has no thickness.
- Both the vector form and the Cartesian form may be used to describe the equations of lines and planes under the angle between a line and a plane.
- These equations may also be used to determine the direction cosines and direction ratios of a line connecting two locations.
- Directional angles are the angles formed by a line and the positive axes of X, Y, and Z.
- If α, β, and γ are a directed line L's directional angles, then cos, cos, and cos are referred to as the line's directional cosines.
Sample Questions
Ques: What are the different types of lines ? (5 marks)
Ans : In geometry there are four types of lines. They are not:
- Horizontal lines : When a line moves from left to right in a straight direction, it is a horizontal line
- Vertical lines : When a line runs from top to bottom in a straight direction, it is a vertical line.
- Parallel lines : When two straight lines don’t meet or intersect at any point, even at infinity, then they are parallel to each other.
Suppose two lines PQ and RS are parallel then it is represented as PQ||RS.
- Perpendicular lines : When two lines meet or intersect at an angle of 90 degrees or at a right angle, then they are perpendicular to each other.
If AB and CD are two lines which are perpendicular to each other, then it is represented as AB ⊥ CD.
Ques:- What is the difference between 2D and 3D geometry? (2 marks)
Ans : Geometry is a study of shapes and figures. Plane geometry or 2D geometry is the type of geometry where there are only 2 dimensions namely length and breadth whereas 3D geometry is the one where the shapes occupy space and have 3 dimensions namely length , width and height.
Ques:- What is the meaning of perpendicular line? (1 mark)
Ans : Perpendicular lines are defined as two lines that meet or intersect each other at right angles (90°).
Ques:- What is the distance formula ? (1 mark)
Ans : d =
Here , d= distance between two points (x1,y1) and (x2,y2)
Ques:- What is the shortest distance between two lines? (1 mark)
Ans : The shortest distance between any two given lines can be expressed as :
The distance between two lines of the form, l1 = a1 + q b1 and l2 = a2 + p b2 is given by the formula :
Ques. Find the equation of plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line 
. (CBSE Delhi 2008)(6 marks)
Ans: The equation passing through (3,4,1) will be : a(x-3) + b(y-4) + c(z-1) = 0 --------------(1)
Since it also passes through (0,1,0) : -3a – 3b – c = 0 ------------(2)
Further, it is parallel to the given line.
Hence, a.2 + b.7 + c.5 = 0 -------------------------(3)
Eliminating a,b and c from equation (2) and (3),
Putting the above values in (1),
-9k(x – 3) – 17k(y-4) – 23k(z-1) = 0
Hence, 9x + 17y + 23z = 20 is the required equation.
Ques. Find the equation of the plane through the line of intersection of the planes 2x + y -z = 3 and 5x - 3y + 4z +9=0 and parallel to the line :
(CBSE AI 2011)(6 marks)
Ans: The equation of the plane passing through the line of intersection of the planes 2x + y -z = 3 and 5x-3y + 4z +9 = 0 is
(2x+y-z-3)+λ(5x-3y + 4z + 9) = 0
x(2 +5λ) + y(1-3λ) + z(4λ-1) +9λ-3=0 ...(i)
Since, plane (i) is parallel to the line
2(2 +5λ) + 4(1-3λ) + 5(4λ-1) +9λ-3=0
λ = -1/6
Putting the value of λ in (i), we obtain
⇒ 7x+9y-10z - 27 = 0
This is the equation of the required plane.
Ques. Find the coordinates of the point where the line through the point A(3, 4, 1) and B(5, 1,6) crosses the XY-plane.(CBSE AI 2012) (4 marks)
Ans: The equation of the line through A(3, 4, 1) and B(5, 1, 6) is
Ques. Find the coordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane, passing through the points (2, 2, 1), (3, 0, 1) and (4, -1,0). (CBSE Delhi 2013)(4 marks)
Ans: Let A(2, 2, 1), B(3, 0, 1) and C(4, -1,0).
Any plane passing through A is given by a(x-2) + b(y-2) + c(z − 1) = 0 …..(i)
As B and C also lie on it, so
a-2b +0 c = 0
2a - 3b - c = 0
From (i), equation of plane is
2(x-2) + (y-2) + (z − 1) = 0
2x+y+z-7=0
Ques. Find the equation of a plane which passes through the point (3, 2, 0) and contains the line 
. (Foreign 2015)(6 marks)
a(x-3) + b(y-2) + c(z - 0) = 0………..(i)
Given line is :
Since plane contains the line, so
a(3-3) + b(6-2) + c(4-0) = 0
0a + 4b + 4c=0 ………..(ii)
and a(1) + b(5) + c(4) = 0
a + 5b + 4c=0 ………..(iii)
Solving (ii) & (iii), we get
a = −4λ, b = 4λ, c = −4λ
Putting values of a, b, c in (i), we get -4λ(x-3) + 4λ(y-2) - 4λ(z - 0) = 0
-4x+12+ 4y - 8-4z = 0
⇒x-y+z-1=0 is the required equation of plane
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