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 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. l^{2} + m^{2} + n^{2} = 1 or cos^{2} α + cos^{2} β + cos^{2} 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 𝑏1⃗⋅𝑏2⃗=0; in cartesian form a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 or l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 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 a_{1}x + b_{1}y + c_{1}z = d_{1} and a_{2}x + b_{2}y + c_{2}z = d_{2}, then equation of any plane passing through the intersection of planes is a_{1}x + b_{1}y + c_{1}z – d_{1} + λ (a_{2}x + b_{2}y + c_{2}z – d_{2}) = 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

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