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The RD Sharma class 12 solution of Directions cosines and Directions ratios exercise 26.1 is one of the most interesting chapters of mathematics Class 12, and students once gain interest in understanding the chapter then they can effortlessly solve these chapters. The Class 12 RD Sharma chapter 26 exercise 26.1 solution will provide you with the best possible ways to solve the questions given and make you understand each and every concept with utmost focus and explanation that are not tough in language to understand.

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Chapter 26- Directions Cosines and Directions Ratios - Ex-FBQ

Chapter 26- Directions Cosines and Directions Ratios - Ex-MCQ

Chapter 26- Directions Cosines and Directions Ratios - Ex-VSA

Given: Angle made by line with x, y and z-axis are

Hint: Find direction cosines of a line

Solution:

Let l, m and n are the direction cosines of the line and are the angles made with axes.

Now,

Therefore, the direction cosines of the line are

Direction Cosines and Direction Ratios exercise 26.1 question 2

Answer:Given: Direction ratios of line are (2,-1,-2)

Hint: Find direction cosines using

Solution:

Let l, m and n are the direction cosines of the line

Here (a, b, c) = (2, -1, -2) are the direction ratios of the line.

Direction cosine is related to direction ratios are

Therefore, direction cosines are

Direction Cosines and Direction Ratios exercise 26.1 question 3

Answer:Hint: Direction ratios are proportional to the length of line

Given: Two points (-2, 4, -5) and (1, 2, 3)

Solution:

The direction ratios of the line joining the two points are

(a, b, c) = (1-(-2), 2-4, 3-(-5))

(a, b, c)=(3, -2, 8)

Now, if l, m and n are the direction cosine, So

Therefore, the direction cosines of the line are

Direction Cosines and Direction Ratios exercise 26.1 question 4

Answer: The points are collinear

Hint: Direction ratios of parallel line are proportional.

Given: Points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11)

Solution:

Direction ratio of AB are = (1-2, -2-3, 3-(-4))

= (-1, -5, -7)

Direction ratio of BC are = (3-1, 8-(-2), -11-3)

= (2, 10, -14)

Comparing direction ratio of AB and BC

Hence, both are proportional and B is the common point in parallel lines.

Therefore, A, B and C are collinear points.

Direction Cosines and Direction Ratios exercise 26.1 question 5

Answer: The direction cosine of the side of triangle ABC areHint: Direction ratios are proportional to the length of side.

Given: A(3, 5, -4), B(-1, 1, 2) and C(-5, -5, 2). Find direction cosine of AB, BC and AC.

Solution: A(3, 5, -4)

B(-1, 1, 2) C(-5, -5, 2)

Let us consider an

Direction ratio of AB = (-1-3, 1-5, 2+4)

= (-4, -4, 6)

Direction cosines of AB are

……………….. (1)

Direction ratio of BC = (-5+1, -5-1, -2-2)

= (-4, -6, -4)

Direction cosines of BC are

………………. (2)

Direction ratio of AC = (-5-3, -5-5, -2+4)

= (-8, -10, 2)

Direction cosines of AC are

………………… (3)

By (1), (2) and (3) we get direction cosine of side AB, BC and AC are

Direction Cosines and Direction Ratios exercise 26.1 question 6

Answer: Angle between two vectors isHint: Use dot product formula

Given: and . Find angle between two vectors.

Solution:

Let be the angle between two vectors with direction ratios &

Now,

Therefore, angle between two vectors is

Direction Cosines and Direction Ratios exercise 26.1 question 7

Answer: Angle between two vectors isHint: Direction cosine is proportional to direction ratios

Given: . Find Angle between two vectors.

Solution:

Let be the angle between two vectors with direction ratios

As direction ratios are proportional to direction cosines

Now,

Therefore, the angle between two vectors is

Direction Cosines and Direction Ratios exercise 26.1 question 8

Answer: Angle between two vectors isHint: Use dot product formula.

Given: . Find angle between two lines

Solution: here we have

Let be the angle between two lines whose direction ratios are &

Therefore the angle between lines is

Direction Cosines and Direction Ratios exercise 26.1 question 9

Answer: The points are collinear.Hint: Direction ratios of parallel lines are proportional.

Given: Points A (2, 3, 4), B (-1, -2, -1), C (5, 8, 7). Prove A, B, C are collinear

Solution: we have Points A (2, 3, 4), B (-1, -2, -1), C (5, 8, 7)

Direction ratio of AB are

Direction ratio of BC are

Comparing direction ratio of AB and BC

Hence, both are proportional and B is the common point in parallel line.

Therefore, A, B, C are collinear points.

Direction Cosines and Direction Ratios exercise 26.1 question 10

Answer: Both lines are parallel to each otherHint: Direction ratio of parallel lines are proportional

Given: . Show that AB and CD are parallel.

Solution: we have

Direction ratio of AB are

Direction ratio of BC are

Comparing direction ratio of AB and BC

Hence, both are proportional to each other

Therefore, AB and CD are parallel lines

Direction Cosines and Direction Ratios exercise 26.1 question 11

Answer: Both the lines are perpendicular to each otherHint: Angle between perpendicular line is

Given: . Show AB is perpendicular to CD

Solution: we have

Let be the angle between two lines whose direction cosines are &

Direction ratio of

Direction ratio of

Now,

Therefore, AB and CD are perpendicular to each other.

Direction Cosines and Direction Ratios exercise 26.1 question 12

Answer: Both the lines are perpendicular to each otherHint: Angle between perpendicular line is

Given: &

. Show AB is perpendicular to CD

Solution: we have &

Let be the angle between two lines whose direction cosines are &

Direction ratio of

Direction ratio of

=

Now,

Therefore, AB and CD are perpendicular to each other.

Direction Cosines and Direction Ratios exercise 26.1 question 13

Answer: Angle between lines isHint: Use dot product formula

Given: . Find angle between two lines.

Solution: we have

Now,

Therefore, the angle between line is

Direction Cosines and Direction Ratios exercise 26.1 question 14

Answer: Angle between AB and CD is

Given: . Find angle between AB and CDSolution: we have

Direction ratio of

Direction ratio of

Let be the angle between two lines whose direction cosines are &

Therefore, angle between AB and CD is

Direction Cosines and Direction Ratios exercise 26.1 question 15

Answer:Given: and . Find direction cosine of line

Solution: we have

………………. (1)

………………. (2)

Given

Or

Thus, direction ratios are proportional to

Now direction cosines are

Direction Cosines and Direction Ratios exercise 26.1 question 16 (i)

Answer: Angle between lines isGiven:

. Find angle between the lines

Hint: Use

Solution: we have

………………. (1)

………………. (2)

Either

Thus, direction ratio of two lines are proportional to

Angle between two lines is

Therefore, angle between two lines is .

Direction Cosines and Direction Ratios exercise 26.1 question 16 (ii)

Answer: angle between two lines is

Given:

, Find angle between the lines

Hint: Use

Solution: we have

…………….. (1)

…………….. (2)

Thus, direction ratio of two lines are proportional to

Angle between two lines is

Therefore, angle between two lines is .

The RD Sharma class 12 solutions chapter 26 exercise 26.1 consists of a total of 19, which covers the essential concepts of this chapter mentioned below-

Direction Cosines

Using direction ratio, show that points are collinear

To find the angle between the lines

Show the points are collinear

Mentioned below are a few reasons why using RD Sharma class 12th exercise 26.1 is beneficial for students of class 12:-

The RD Sharma class 12 chapter 26 exercise 26.1 solutions are designed by professionals who are experts in this field and thus provide you with the best content that cant be found in any other study material .

These solutions are trusted by thousands of students and teachers as it has helped them both because teachers use it for giving lectures and preparing question papers and students get profit by scoring high marks.

The RD Sharma class 12th exercise 26.1 is also helpful in solving homework as it contains solved questions for reference and also saves time.

Students should make a habit to go through these solutions regularly and revise it so as to master the subject of maths and make a remarkable performance in the board exams.

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- Chapter 24 - Vector or Cross Product
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1. How do you find the direction ratio of direction cosines?

Any numbers that are proportional to the direction cosines are called direction ratios usually represented as a, b, c. So we can write, a=kl, b=km, c=kn where k is a constant.

2. What is the difference between direction cosine and direction ratios?

We can say that cosines of direction angles of a vector are the coefficients of the unit vectors and when the unit vector is resolved in terms of its rectangular components.

3. Is this solution helpful for the preparation of public exams?

Yes, the RD Sharma class 12 solution is helpful for the preparation of public exams as well as the board exams.

4. How do you find the direction of A ratio?

To find the components of a unit vector, divide the original three components of the vector by the magnitude of the vector. the three components of the unit vector are known as direction ratios because they represent the ratio of each coordinate to the total magnitude.

5. Do parallel lines have the same direction ratios?

Yes, parallel lines do have the same direction ratios.

Mar 22, 2023

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