ML Aggarwal Three Dimensional Geometry ISC Class-12 Maths Solutions

ML Aggarwal Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B. Step by step Solutions of ML Aggarwal ISC Class 12 Mathematics for Exercise Questions with Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

ML Aggarwal Three Dimensional Geometry ISC Class-12 Maths Solutions

ML Aggarwal Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B

Board   ISC
Class  12
Subject Mathematics
Chapter-2 Three Dimensional Geometry
Session  2024-25
Topics  Solutions of ML Aggarwal

Three Dimensional Geometry

Direction Cosines of a Line: If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis and Z-axis respectively, then cos α, cos β, and cos γ, are called direction cosines of a line. They are denoted by l, m, and n. Therefore, l = cos α, m = cos β and n = cos γ. Also, sum of squares of direction cosines of a line is always 1,
i.e. l2 + m2 + n2 = 1 or cos2 α + cos2 β + cos2 γ = 1
Note: Direction cosines of a directed line are unique.

Straight line: A straight line is a curve, such that all the points on the line segment joining any two points of it lies on it.

Condition of Perpendicularity:

Two lines are said to be perpendicular, when in vector form 𝑏1⃗𝑏2⃗=0; in cartesian form a1a2 + b1b2 + c1c2 = 0  or l1l2 + m1m2 + n1n2 = 0 [direction cosine form]

Plane:

A plane is a surface such that a line segment joining any two points of it lies wholly on it. A straight line which is perpendicular to every line lying on a plane is called a normal to the plane.

Equations of a Plane in Normal form

Vector form: The equation of plane in normal form is given by 𝑟⃗ 𝑛⃗ =𝑑, where 𝑛⃗  is a vector which is normal to the plane.
Cartesian form: The equation of the plane is given by ax + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from origin.
Another equation of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from origin and p is a distance of a plane from origin.
Note: If d is the distance from the origin and l, m and n are the direction cosines of the normal to the plane through the origin, then the foot of the perpendicular is (ld, md, nd).

Cartesian form:

If the equation of planes are a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2, then equation of any plane passing through the intersection of planes is a1x + b1y + c1z – d1 + λ (a2x + b2y + c2z – d2) = 0  where, λ is a constant and calculated from given condition.


Exercise – 2.1

ML Aggarwal Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B


Exercise – 2.2

Three Dimensional Geometry ISC Class-12 Maths Solutions


Exercise – 2.3

Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B


Exercise – 2.4

ML Aggarwal ISC Class-12 Maths Solutions


Exercise – 2.5

ML Aggarwal Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B


Exercise – 2.6

Three Dimensional Geometry ISC Class-12 Maths Solutions


Exercise – 2.7

Three Dimensional Geometry ISC Class-12 Maths Solutions


Exercise – 2.8

 Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B


Chapter Test

ML Aggarwal Three Dimensional Geometry

-: End of Three Dimensional Geometry ISC Class-12 Maths Solutions Chapter-2 Section-B :-

Return to :-  ML Aggarwal ISC Class 12 Maths Vol-2 Solutions

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