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

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. 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.

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 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:

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.

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 :-

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### 4 thoughts on “OP Malhotra Three Dimensional Geometry ISC Class-12 Maths Solutions Ch-23”

1. Sir isme solutions toh digiye please