Class 12 RD Sharma Solutions - Chapter 24 Scalar or Dot Product - Exercise 24.1 | Set 2

Question 17. If and , then express  in the form of where is parallel to and is perpendicular to .  

Solution:

Given, 

According to question

 also  = 0

Now,

⇒ 

⇒ 

⇒ 

Now, 

⇒ 

⇒ 3(2-3λ)+4(1-4λ)-5(4+5λ) = 0

⇒ 6-9λ+4-16λ-20-25λ = 0

⇒ -10 -50λ = 0

⇒ λ = -1/5

Question 18. If either or , then . But, The converse need not be true. Justify your answer with an example.

Solution:

Given, 

 or  then 

Suppose 

But,

= √(2)²+(1)²+(1)²

= √4+1+1

= √6 ≠ 0

= √(1)²+(1)²+(1)²

= √3 ≠ 0

Hence Proved

Question 19. Show that the vectors  form a right-angled triangle.

Solution:

Given, 

To prove given vectors form a right angle triangle

= √(3²+(-2)²+1²) = √14

= √(1²+(-3)²+5²) = √35

= √(2²+1²+(-4)²) = √21

= 14 + 21 = 35

Since, (Pythagoras Theorem)

Hence, and  form a right angled triangle.

Question 20. If , and are such that is perpendicular to , then find the value of λ.

Solution:

Given:

Now, 

⇒ 

⇒ 

⇒ (2 - λ)3 + (2 + 2λ) + 0 = 0

⇒ 6 - 3λ + 2 + 2λ =0

⇒ λ = 8

Question 21.  Find the angles of a triangle whose vertices are A (0, -1. -2), B (3, 1, 4), and C(5, 7, 1).

Solution:

Given that angle of a triangle whose vertices are A (0, -1. -2), B (3, 1, 4) and C(5, 7, 1).

= √98 = 7√2

Now, 

 = (3 × 2 + 2 × 6 - 6 × 3) = 0

Thus, we can say AB is perpendicular to BC.

Hence, AB = BC = 7, ∠A =∠C and ∠B = 90°

∠A + ∠B + ∠C = 180°

2∠A = 180° - 90°

∠A = 45°

∠C = 45°  

∠B = 90°

Question 22. Find the magnitude of two vectors and , having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.

Solution:

We know 

⇒ 1/ 2 = 

⇒ 1/2 = (1/2)

⇒ 

or

⇒ 

Question 23. Show that the points whose position vector are  form a right triangle.

Solution:

Given that positions vectors

Now,

⇒ 

⇒ 

⇒ 

Now, 

= 2 - 3 - 20 = -21

= -3 - 6 - 5 = -14

= -6 + 2 + 4 = 0

So, AB is perpendicular to CA or the given position vectors form a right-angled triangle.

Question 24. If the vertices A, B, C of △ABC have position vectors (1, 2, 3), (-1, 0, 0), (0, 1, 2) respectively, what is the magnitude of ∠ABC?

Solution:

Given the vertices of △ABC are A(1, 2, 3), B(-1, 0, 0), C(0, 1, 2)

Now, 

= 

Or, 

We know that 

 (2 × 1) + (2 × 1) + (3 × 2)

= 2 + 2 + 6 = 10

Now,  = √17                          

 = √6

Therefore,

 cos θ = 

 cos θ = 10/ √(17×6)

θ = cos^-1(10/√102)                 

Question 25. If A, B, C have position vectors (0, 1, 1), (3, 1, 5), (0, 3, 3) respectively, show that △ABC is right-angled at C. 

Solution:

Given, position vectors A(0, 1, 1), B(3, 1, 5), C(0, 3, 3)

Now, 

= 

= 2 × 2 - 2 × 2 = 0

Thus,  and  are perpendicular hence △ABC is right-angled at C

Question 26. Find the projection of on , where and.

Solution:

Given: 

To find the projection of  on 

Now, Projection of = 

= 

= 6/9 × 3

= 2

Question 27. If and , then show that the vectors and  are orthogonal.

Solution:

Given: 

To prove 

Taking LHS

= 

= 

= √35 - √35

= 0 

Thus, the given vectors and are orthogonal.

Question 28. A unit vector makes angle π/2 and π/3 with and respectively and an acute angle θ with . Find the angle θ and components of .

Solution:

Let us assume 

We know that 

a₁²+ a₂²+ a₃² = 1  ....(1)

So, 

(1)(1)(1/√2) = a₁

a₁ = 1/√2

Again we take 

(1)(1)(1/2) = a₂

a₂ = 1/2

Put all these values in eq(1) to find the value of a₃

(1/√2)²+ (1/2)²+ a₃² = 1  ....(1)

a₃² = 1/4

a₃ = 1/2

Now we find the value of θ 

(1)(1)cosθ = 1/2

cosθ = 1/2

cosθ = π/3

and components of 

Question 29. If two vectors and are such that = 2, = 1, and =1. Find the value of 

Solution:

Given, 

=

=

= 6(2)² + 11(1) - 35(1)²

= 24 + 11 - 35

= 35 - 35 = 0

Question 30. If is a unit vector, then find in each of the following:

(i) 

Solution:

Given, 

⇒

⇒ 

⇒ 

⇒ 

⇒

(ii) 

Solution:

Given, 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ =√13

Question 31.  Find and , if  

(i) = 12 and 

Solution:

Given,  = 12

⇒ 

⇒  = 12

⇒  = 12

⇒ = 12

⇒ = 2

So, 

= 4

(ii) = 8 and = 8

Solution:

Given, = 8

⇒ 

⇒ 

⇒

⇒ 

⇒ = √(8/63)

So, 

= 8√(8/63)

(iii)= 3 and  = 2

Solution:

Given, 

⇒ 

⇒ 

⇒ 

⇒ 3= 3

⇒ = 1

So, 

= 2

Question 32. Find , if  

(i) and 

Solution:

We have, 

⇒

⇒= 2² - 2 × 8 + 5²

⇒ = 4 - 16 + 25

⇒ = 13

⇒= √13

(ii) = 3, = 4 and  = 1

Solution:

We have, 

⇒ 

⇒ = 3² - 2 × 1 + 4²

⇒ = 9 - 2 + 16

⇒ = 23

⇒ = √23

(iii)  and = 4

Solution:

We have, 

⇒ 

⇒= 2² - 2 × 4 + 3²

⇒ = 4 - 8 + 9

⇒ = 5

⇒ = √5

Practice Problems on Scalar or Dot Product

1. Find the scalar product of vectors a = 2i + 3j - k and b = i - 2j + 4k.

2. If a = 3i - 2j + k and b = 2i + 4j - 3k, calculate a · b.

3. Determine if vectors a = i + 2j + 3k and b = 2i - j + k are perpendicular.

4. Find the angle between vectors a = 2i + 2j + k and b = i + j + 2k.

5. Calculate the projection of vector a = 3i - 4j + 2k onto vector b = i + 2j - 2k.

6. If a · b = 10 and |a| = 3, |b| = 5, find the angle between a and b.

7. Prove that (a + b) · (a - b) = |a|² - |b|².

8. Find a unit vector perpendicular to both a = i + j + k and b = 2i - j + 2k.

9. If a = 2i - j + 3k and b = i + 2j - k, find a scalar λ such that (a + λb) · b = 0.

10. Determine the value of x if (2i + xj - k) · (i + 2j + 3k) = 10.

Summary

Chapter 24 of RD Sharma Class 12 Solutions focuses on the Scalar or Dot Product of vectors. Key concepts covered include:

• Definition of scalar product
• Properties of scalar product
• Geometric interpretation of scalar product
• Applications in finding angles between vectors
• Projections of vectors
• Conditions for perpendicularity and parallelism of vectors
• Vector equations using scalar product

The chapter emphasizes the importance of scalar product in vector algebra and its applications in physics and engineering.