RD Sharma Solutions for Class 12 Maths Chapter 23 – Algebra of Vectors

Explore comprehensive exercise-wise RD Sharma Solutions for Class 12 Chapter 23 – Algebra of Vectors, carefully prepared to align with the latest CBSE syllabus and examination patterns. These solutions are designed by experienced educators to help Class 12 students grasp key vector concepts and solve textbook problems with clarity and accuracy.

For students gearing up for the Class 12 Maths board exam, regular practice is essential. Selecting the right study resources plays a vital role in mastering challenging topics. To support students in this journey, our RD Sharma solutions offer step-by-step answers to every exercise question in the textbook. You can view these solutions online or download them as a PDF to study anytime, anywhere—even without internet access.

RD Sharma Solution for Class 12 Chapter 23 - Download PDF

Here are the RD Sharma Solutions Class 9 Maths Chapter 25 Probability Solutions, designed to help students prepare effectively for their exams. By referring to these solutions and practicing the problems, students can boost their confidence and improve their scores.

RD Sharma Solution for Class 12 Chapter 23 - Question with Answers

Question 1 Let vector a = i hat + 2 j hat + 3 k hat and vector b = 4 i hat - j hat + k hat. Find a + b.

Solution:
a + b = (1 + 4) i hat + (2 - 1) j hat + (3 + 1) k hat = 5 i hat + 1 j hat + 4 k hat

Question 2 Find the vector difference a - b where a = 2 i hat + 3 j hat and b = i hat - 4 j hat.

Solution:
a - b = (2 - 1) i hat + (3 + 4) j hat = 1 i hat + 7 j hat

Question 3 If vector a = 3 i hat + 4 j hat, find the magnitude of a.

Solution: Magnitude = square root of (3 squared + 4 squared) = square root of (9 + 16) = square root of 25 = 5

Question 4 Let vector a = i hat + 2 j hat and vector b = 2 i hat - j hat. Find the scalar multiplication 3a.

Solution: 3a = 3 times (1 i hat + 2 j hat) = 3 i hat + 6 j hat

Question 5 Given a = i hat + j hat and b = j hat + k hat, find a + 2b.

Solution: 2b = 2 j hat + 2 k hat
a + 2b = i hat + j hat + 2 j hat + 2 k hat = i hat + 3 j hat + 2 k hat

Question 6 If vector a has magnitude 6 and direction along unit vector i hat, what is the vector a?

Solution:
a = 6 i hat

Question 7 Find the unit vector in the direction of a = 3 i hat + 4 j hat.

Solution:
Magnitude = 5
Unit vector = (3 divided by 5) i hat + (4 divided by 5) j hat

Question 8 If a = 2 i hat + 3 j hat + 4 k hat and b = i hat - j hat + k hat, find the scalar product a dot b.

Solution: = 21 + 3(-1) + 4*1 = 2 - 3 + 4 = 3

Question 9 If a = i hat + j hat, find the vector perpendicular to a in 2D.

Solution: A vector perpendicular to a is -j hat + i hat or -i hat + j hat (any orthogonal rotation)

Question 10 If a = i hat + j hat + k hat, find the magnitude of 2a.

Solution:
2a = 2 i hat + 2 j hat + 2 k hat
Magnitude = square root of (4 + 4 + 4) = square root of 12

Question 11 Find vector x such that x + a = b, where a = i hat + j hat and b = 2 i hat + 3 j hat

Solution:
x = b - a = (2 - 1) i hat + (3 - 1) j hat = i hat + 2 j hat

Question 12 If vector a and b are equal, and a = 3 i hat + x j hat, b = 3 i hat + 2 j hat, find x.

Solution:
x = 2

Question 13 If a = 2 i hat + 3 j hat and b = 4 i hat - j hat, find a - 2b.

Solution:
2b = 8 i hat - 2 j hat
a - 2b = (2 - 8) i hat + (3 + 2) j hat = -6 i hat + 5 j hat

Question 14 Find a vector of magnitude 10 in the direction of 3 i hat + 4 j hat.

Solution:
Unit vector = (3/5) i hat + (4/5) j hat
Required vector = 10 * unit vector = 6 i hat + 8 j h

Question 15 Find the magnitude of a = 7 i hat + 24 j hat.

Solution: Magnitude = square root of (49 + 576) = square root of 625 = 25

Question 16 Given a = i hat + j hat, find the vector b such that a + b is along the k hat direction.

Solution:
To cancel i and j components, b = -i hat - j hat
Then a + b = 0 i hat + 0 j hat + 0 k hat (null vector, or no i and j parts)

Question 17 Find a vector whose magnitude is 1 and direction is same as that of 6 i hat - 8 j hat.

Solution:
Magnitude = 10
Unit vector = (6/10) i hat - (8/10) j hat = 0.6 i hat - 0.8 j hat

Question 18 Find sum of vectors a = 2 i hat and b = -3 i hat + 4 j hat

Solution:
a + b = (2 - 3) i hat + 4 j hat = -1 i hat + 4 j hat

Question 19 Let a = i hat + j hat + k hat, find a + a + a.

Solution:
3a = 3 i hat + 3 j hat + 3 k hat

Question 20 If vector a = i hat + j hat, find a dot a.

Solution:
= 1 squared + 1 squared = 2

Question 21 Find the vector whose magnitude is square root of 2 and is equally inclined to i hat and j hat.

Solution:
Unit vector = (1 divided by square root of 2) i hat + (1 divided by square root of 2) j hat

Question 22 Given a = i hat - j hat + k hat and b = 2 i hat + j hat - k hat, find a + b

Solution:
= (1 + 2) i hat + (-1 + 1) j hat + (1 - 1) k hat = 3 i hat + 0 j hat + 0 k hat

Question 23 If a vector makes an angle of 45 degrees with both x and y axes in 2D, find its unit vector.

Solution:
= (1 divided by square root of 2) i hat + (1 divided by square root of 2) j hat

Question 24 Find projection of a = 3 i hat + 4 j hat on b = i hat

Solution:
Projection = (31 + 40) divided by magnitude of b = 3

Question 25 If vector a is perpendicular to vector b, then what is a dot b?

Solution: a dot b = 0 (since dot product of perpendicular vectors is zero)