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NCERT Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry are provided in this article. A three-dimensional shape, in Geometry, can be described as a solid figure or an object or shape that has three dimensions – length, width, and height. As compared to two-dimensional shapes, three-dimensional shapes have thickness or depth.
Chapter 11 will carry a total weightage of 7 to 14 marks in the CBSE Class 12 examination. Around 3-4 short as well as long answer questions can come from Angle between two lines, Plane, Angle between line and plane, Angle between two vectors, Coplanarity, Angle between a line and a plane.
Download PDF: NCERT Solutions for Class 12 Mathematics Chapter 11
Class 12 Maths NCERT Solutions Chapter 11 Three Dimensional Geometry
NCERT Solutions Class 12 Mathematics Chapter 11 Important Topics
Chapter 11 Three Dimensional Geometry is an important topic in the board examination as per CBSE Class 12 exam pattern. In NCERT Class 12 Mathematics Chapter 11, direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, the shortest distance between two lines, Cartesian and vector equation of a plane, distance of a point from a plane are discussed.
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Angle between Two Lines
The angle between two lines, which is the measure of the inclination between the two lines, enables us to know the relationship between the two lines. The angle between two lines will be θ = Tan-1m, of which one of the lines is y = mx + c, and the other line is the x-axis. The angle between two lines that are parallel to each other and have equal slopes (m1=m2 m 1 = m 2 ) is 0º.
Angle between Two Lines
| The formulas for Angle between Two Lines are:
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Plane
A plane, in three-dimensional geometry, can be defined by the three points it contains, as long as those points are not on the same line. It is the two-dimensional analog of a point (zero dimensions), a line (one dimension), and a three-dimensional space.
Plane
| A plane in three-dimensional space can be represented as: ax + by + cz + d + 0 where at least one of the numbers a, b, and c must be non-zero. A plane in 3D coordinate space is determined by a point and a vector that is perpendicular to the plane. |
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Angle between a Line and a Plane
When a line is inclined on a plane, and a normal is drawn to the plane from a point where it is touched by the line an angle between a line and a plane is formed. This angle between a line and a plane is equal to the complement of an angle between the normal and the line.
| The two ways of calculating the angle between a plane and a straight line are: Cartesian Form The straight line equation in Cartesian form is defined 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| |
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Angle between Two Vectors
The angle between two vectors is defined as the angle between their tails. The angle can be found either by using the dot product (scalar product) or the cross product (vector product). It must be noted that the angle between two vectors always lies between 0° and 180°.
Angle between two vectors
| The angle between two vectors a and b is found using the formula: θ = cos-1 [ (a · b) / (|a| |b|) ]. If the two vectors are equal, then substitute b = a in this formula, then we get θ = cos-1 [ (a · a) / (|a| |a|) ] = cos-1 (|a|2/|a|2) = cos-11 = 0°. |
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Coplanarity
Coplanar lines, which are a common concept with regard to 3-dimensional geometry, are referred to as the lines that lie on the same plane. The two lines are coplanar can be proven by using the condition in vector form and Cartesian form.
Coplanar Lines
| Conditions for Coplanarity Using Cartesian form: The condition for coplanarity in the Cartesian form emerges from the vector form. The vector form of equation of the line connecting L and Q can be given as under: LQ = (a2 – a1)i + (b2 – b1)j + (c2 – c1)k Q1 = x1i + y1j + z1k Q2 = x2 i + y2 j + z2 k Condition for Coplanarity Using Vector form: For the derivation of the condition for coplanarity in vector form, we shall consider the equations of two straight lines to be as below: r1 = l1 + λq1 r2 = l2 + λq2 |
NCERT Solutions For Class 12 Maths Chapter 11 Exercises
- Exercise 11.1 Solutions
- Exercise 11.2 Solutions
- Exercise 11.3 Solutions
- Miscellaneous Exercise Solutions
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