#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

## Chapter 27: Direction Cosines and Direction Ratios

#### Chapter 27: Direction Cosines and Direction Ratios solutions [Pages 22 - 23]

If a line makes angles of 90°, 60° and 30° with the positive direction of *x*, *y*, and *z*-axis respectively, find its direction cosines

If a line has direction ratios 2, −1, −2, determine its direction cosines.

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .

Using direction ratios show that the points *A* (2, 3, −4), *B* (1, −2, 3) and *C* (3, 8, −11) are collinear.

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).

Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.

Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).

Find the angle between the lines whose direction ratios are proportional to *a*, *b*, *c* and *b* − *c*, *c* − *a*, *a*− *b*.

If the coordinates of the points *A*, *B*, *C*, *D* are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between *AB* and *CD*.

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.

Find the angle between the lines whose direction cosines are given by the equations

(i) *l *+ *m* + *n* = 0 and *l*^{2} + *m*^{2} − *n*^{2} = 0

Find the angle between the lines whose direction cosines are given by the equations

2*l* − *m* + 2*n* = 0 and *mn* + *nl* + *lm* = 0

Find the angle between the lines whose direction cosines are given by the equations

*l* + 2*m* + 3*n* = 0 and 3*lm* − 4*ln* + *mn* = 0

Find the angle between the lines whose direction cosines are given by the equations

2*l* + 2*m* − *n* = 0, *mn* + *ln* + *lm* = 0

#### Chapter 27: Direction Cosines and Direction Ratios solutions [Pages 24 - 25]

Define direction cosines of a directed line.

What are the direction cosines of X-axis?

What are the direction cosines of *Y*-axis?

What are the direction cosines of *Z*-axis?

Write the distances of the point (7, −2, 3) from *XY*, *YZ* and *XZ*-planes.

Write the distance of the point (3, −5, 12) from *X*-axis?

Write the ratio in which *YZ*-plane divides the segment joining *P* (−2, 5, 9) and *Q* (3, −2, 4).

A line makes an angle of 60° with each of *X*-axis and *Y*-axis. Find the acute angle made by the line with *Z*-axis.

If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.

Write the ratio in which the line segment joining (*a*, *b*, *c*) and (−*a*, −*c*, −*b*) is divided by the *xy*-plane.

Write the inclination of a line with *Z*-axis, if its direction ratios are proportional to 0, 1, −1.

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

Write the distance of the point *P* (*x*, *y*, *z*) from *XOY* plane.

Write the coordinates of the projection of point *P* (*x*, *y*, *z*) on *XOZ*-plane.

Write the coordinates of the projection of the point *P* (2, −3, 5) on *Y*-axis.

Find the distance of the point (2, 3, 4) from the *x*-axis.

If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?

Write direction cosines of a line parallel to *z*-axis.

If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.

Answer each of the following questions in one word or one sentence or as per exact requirement of the question:

Write the distance of a point P(*a*, *b*, *c*) from *x*-axis.

If a line makes angles 90° and 60° respectively with the positive directions of *x* and *y* axes, find the angle which it makes with the positive direction of *z*-axis.

#### Chapter 27: Direction Cosines and Direction Ratios solutions [Pages 25 - 26]

For every point *P* (*x*, *y*, *z*) on the *xy*-plane,

*x*= 0*y*= 0*z*= 0*x*=*y*=*z*= 0

For every point *P* (*x*, *y*, *z*) on the *x*-axis (except the origin),

*x*= 0,*y*= 0,*z*≠ 0*x*= 0,*z*= 0,*y*≠ 0*y*= 0,*z*= 0,*x*≠ 0*x*=*y*=*z*= 0

A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is

2

3

4

all of these

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is

7

`sqrt(38)`

`sqrt(155)`

none of these

The *xy*-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)

internally in the ratio 2 : 3

externally in the ratio 2 : 3

internally in the ratio 3 : 2

externally in the ratio 3 : 2

If the *x*-coordinate of a point *P* on the join of *Q* (2, 2, 1) and *R* (5, 1, −2) is 4, then its z-coordinate is

2

1

-1

-2

The distance of the point *P* (*a*, *b*, *c*) from the *x*-axis is

\[\sqrt{b^2 + c^2}\]

\[\sqrt{a^2 + c^2}\]

\[\sqrt{a^2 + b^2}\]

none of these

Ratio in which the *xy*-plane divides the join of (1, 2, 3) and (4, 2, 1) is

3 : 1 internally

3 : 1 externally

1 : 2 internally

2 : 1 externally

If *P* (3, 2, −4), *Q* (5, 4, −6) and *R* (9, 8, −10) are collinear, then *R* divides *PQ* in the ratio

3 : 2 externally

3 : 2 internally

2 : 1 internally

2 : 1 externally

If *O* is the origin, *OP* = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of *P* are

(−1, 2, −2)

(1, 2, 2)

(−1/9, 2/9, −2/9)

(3, 6, −9)

The angle between the two diagonals of a cube is

(a) 30°

(b) 45°

(c) \[\cos^{- 1} \left( \frac{1}{\sqrt{3}} \right)\]

(d) \[\cos^{- 1} \left( \frac{1}{3} \right)\]

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos^{2} α + cos^{2} β + cos^{2}γ + cos^{2} δ is equal to

\[\frac{1}{3}\]

\[\frac{2}{3}\]

\[\frac{4}{3}\]

\[\frac{8}{3}\]

## Chapter 27: Direction Cosines and Direction Ratios

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

#### Textbook solutions for Class 12

## RD Sharma solutions for Class 12 Mathematics chapter 27 - Direction Cosines and Direction Ratios

RD Sharma solutions for Class 12 Maths chapter 27 (Direction Cosines and Direction Ratios) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 27 Direction Cosines and Direction Ratios are Direction Cosines and Direction Ratios of a Line, Equation of a Line in Space, Shortest Distance Between Two Lines, Plane Passing Through the Intersection of Two Given Planes, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Equation of a Plane in Normal Form, Three - Dimensional Geometry Examples and Solutions, Introduction of Three Dimensional Geometry, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Angle Between Two Lines, Vector and Cartesian Equation of a Plane.

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