OP Malhotra **Three Dimensional Geometry** ISC Class-12 Maths Solutions Ch-23. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-23(a), Exe-23(b), Exe-23(c), Exe-23(d), Exe-23(e), and Exe-23(f) . Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

## OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Class: | 12th |

Board | ISC |

Subject: | Mathematics |

Chapter : | Ch-23 Three Dimensional Geometry of Section -B |

Topics | Solutions of Exe-23(a), Exe-23(b), Exe-23(c), Exe-23(d), Exe-23(e), Exe-23(f) . |

Writer | OP Malhotra, SK Gupta, Anubhuti Gangal |

Publications | S.Chand Publications 2020-21 |

## OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

**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.

**Equation of Line Passing through Two Given Points:**

**Vector form:** 𝑟⃗ =𝑎⃗ +𝜆(𝑏⃗ −𝑎⃗ ), λ ∈ R, where a and b are the position vectors of the points through which the line is passing.

**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:**

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.

**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).

**Exe-23(a)**

### OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Question 1: The direction ratios of a line are -1, -2, -2. What are their direction cosines ?

Question 2: If α, β, γ are angle s which a line makes with the axes, prove that sin² α + sin² β + sin² γ = 2.

Question 3: Can a line have direction angles 45, 60, 120 degrees ?

Question 4: ……………..

Question 27: Find the angle between any two diagonals of a cube.

**Exe-23(b)**

### OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Question 1: Passing through the point (-, 2, 3) and having direction ratios proportional to -4, 5, 6.

Question 2: Passing through the point (2, -3, 0) and having direction cosines -1/7, 4/7, -6/7

Question 3: Passing through the point (2, 3, 4) and (4, 6, 5).

Question 4: ……………….

Question 15: The equation of a line is ……………

Find the direction cosines of a line parallel to the line.

**Exe-23(c)**

### OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Question 1: Find the vector equation of a line which passes ………………. Find the Cartesian from also.

Question 2: Find the vector equation of a line which is parallel to the vector …………………… it to the Cartesian from.

Question 3: Find the vector and Cartesian equation of the line that passes …………. (3, -2, 6)

Question 4: …………………

Question 10: Find the direction cosine and vector equation of the line whose Cartesian from is …………………… = -1.

**Exe-23(d)**

OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Question 1: Find the angle between the following pairs of lines :

(i) ……………………………

Question 2: Find the angle between the following pairs of lines :

(i) …………………..

Question 3: Find the angle between the pairs of lines with direction ratios :

(i) 2, 2, 1 and 4, 1, 8 (ii) 1, 2, -2 and -2, 2, 1.

Question 4: Prove that the lines ……………….. right angle

Question 5: ……………………..

Question 18: Find the image of the point (2, -1, 5) in the line ……..

**Exe-23(e)**

OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Question 1: Show that the line ………….. of intersection.

Question 2: Prove that the lines ……………… point of intersection.

Question 3: Show that the line ……………………… are coplanar.

Question 4: Show that the line …………………….. do not intersect each other

Question 5: Show that the line ……………………… are coplanar.

Question 6: Find the equations of the line which intersects the lines ………… and passes through (1, 1, 1).

**Exe-23(f)**

### OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23

Find the length of the shortest distance between the lines.

Question 1: (x – 3)/ 1= (y – 5)/-2 = (z – 7)/1 and (x + 1)/7 = (y + 1)/-6 = (z + 1)/1

Question 2: ………………..

Question 10: Find the shortest distance between the following pairs of parallel lines.

(i)………………..

Question 11: Define the line of shortest distance between two skew lines. Find the shortest distance and the vectot ……….. lines given by :

(i)…………………..

**-: End of Three Dimensional Geometry ****S. Chand ISC Class-12 Maths Solution :-**

Return to :- OP Malhotra S. Chand ISC Class-12 Maths Solutions

Thanks

Please share with your friends

Sir isme solutions toh digiye please

yes given now

Sir anwer dikhai nhi rha hai refuse content dikha rha hai

our engineer is suffering from dengue currently. even your problem will be solve soon