# NCERT solutions for class 12 Maths Chapter 10

NCERT solutions for class 12 Maths Chapter 10 Vector Algebra exercise 10.1, 10.2, 10.3, 10.4 & miscellaneous exercises in PDF form for UP Board and CBSE Board students to free download for 2020-21.

Download Class 12 solutions all subjects, Solutions are prepared according to CBSE Syllabus for 2020-2021. Visit to Discussion Forum to share your knowledge.## NCERT solutions for class 12 Maths Chapter 10

Class: | 12 |

Subject: | Maths |

Chapter 10: | Vector Algebra |

### 12th Maths Chapter 10 Solutions

NCERT solutions for class 12 Maths Chapter 10 in PDF form to free download for academic session 2020-21. Join the discussion forum to ask your questions related to NIOS and CBSE Board, and reply to the questions asked by others. Download NCERT Books 2020-2021 based on latest CBSE Syllabus.

### Chapter 10 Solutions in PDF

- Download Exercise 10.1 in PDF
- Download Exercise 10.2 in PDF
- Download Exercise 10.3 in PDF
- Download Exercise 10.4 in PDF
- Download Miscellaneous Exercise 10
- NCERT Book Chapter 10
- NCERT Book Answers
- Revision Book Chapter 10
- Revision Book Answers
- Download Assignment 1
- Download Assignment 2
- Download Assignment 2 Answers
- Download Assignment 3
- Download Assignment 4
- Visit to 12th Maths Main Page

### Solutions of Assignment given below

#### Important Questions Assignment with Solutions

Q.1. If AB = 3i + 2j – k and the coordinates of A are (4, 1, 1), then find the coordinates of B.

Q.2. Let a = -2i + j, b = i + 2j and c = 4i + 3j. Find the values of x and y such that c = xa + yb.

Q.3. Find a unit vector in the direction of the resultant of the vectors i – j+ 3k, 2i + j – 2k and i + 2j – 2k.

Q.4. Find a vector of magnitude of 5 units parallel to the resultant of vector a = 2i + 3j + k and b = i – 2j – k.

Q.5. For what value λ are the vectors a and b perpendicular to each other? Where a = λi + 2j + k and b = 5i – 9j + 2k.

Q.6. Write the value of p for which a = 3i + 2j + 9k and b = i + pj + 3k are parallel vectors.

Q.7. For any two vectors a and b, write when |a + b|=|a – b| holds.

Q.8. Find the value of p if (2i + 6j + 27k)×(i + 3j + pk) = 0.

Q.9. Evaluate: i.(j × k) + (i × k).j.

Q.10. If a = 2i – 3j, b = i + j – k, c = 3i – k, find [a b c].

Q.11. If a = 5i – 4j + k, b = -4i + 3j – 2k and c = i – 2j – 2k, then evaluate c.(a × b).

Q.12. Show that vectors i + 3j + k, 2i – j – k, 7j + 3k are parallel to same plane.

Q.13. Find a vector of magnitude 6 which is perpendicular to both the vectors 2i – j + 2k and 4i – j + 3k.

Q.14. If a.b = 0, then what can you say about a and b?

Q.15. If a and b are two vectors such that |a × b| = a.b, then what is the angle between a and b?

Q.16. Find the area of a parallelogram having diagonals 3i + j – 2k and i – 3j + 4k.

Q.17. If i, j and k are three mutually perpendicular vectors, then find the value of j.(k × i).

Q.18. P and Q are two points with position vectors 3a – 2b and a + b respectively. Write the position vector of a point R which divides the segment PQ in the ratio 2∶1 externally.

Q.19. Find λ when scalar projection of a = λi + j + 4k on b = 2i + 6j + 3k is 4 units.

Q.20. Find “a” so that the vectors p = 3i – 2j and q = 2i – aj be orthogonal.

Q.21. If a = i – j + k, b = 2i + j – k and c = λi – j + λk are co-planar, find the value of λ.

Q.22. What is the point of trisection of PQ nearer to P if positions of P and Q are 3i + 3j – 4k and 9i + 8j – 10k respectively?

Q.23. What is the angle between a and b, if a.b = 3 and |a × b| = 3√3.

##### Short Answer type Questions (2 Marks)

Q.1. A vector r is inclined to x- axis at 45° and y- axis at 60° if |r| = 8 units find r.

Q.2. If |a + b|= 60, |a – b| = 40 and b = 46 find |a|.

Q.3. Write the projection of b + c on a where a = 2i – 2j + k, b = i + 2j – 2k and c = 2i – j + 4k.

Q.4. If the points (-1, -1, 2), (2, m, 5) and (3, 11, 6) are co-linear, find the value of m.

Q.5. For any three vectors a, b and c write value of the following. a × ( b + c ) + b × (c + a ) + c × (a + b).

Q.6. If (a + b)² + (a . b)² = 144 and |a|, then find the value of |b|.

Q.7. If for any two vectors a and b, (a + b)² + (a – b)² = λ[a² + b²] then write the value of λ.

Q.8. If a, b are two vectors such that |a + b| = |a| then prove that 2a + b is perpendicular to b.

Q.9. Show that vectors a = 3i – 2j + k, b = i – 3j + 5k, c = 2i + j – 4k form a right angle triangle.

Q.10. If a, b, c are three vectors such that a + b + c = 0 and |a| = 5, |b| = 12, |c| = 13, then find a.b + b.c + c.a

Q.11. The two vectors i + j and 3i – j + 4k represents the two sides AB and AC respectively of ΔABC, find the length of median through A.

###### Short Answer type Questions (4 Marks)

Q.1. The points A, B and C with position vectors 3i – yj + 2k, 5i – j + k and 3xi + 3j – k are collinear. Find the values of x and y and also the ratio in which the point B divides AC.

Q.2. If the sum of two unit vector is a unit vector, prove that the magnitude of their difference is √3.

Q.3. Let a = 4i + 5j – k , b = i – 4j + 5k and c = 3i + j – k. Find a vector d which is perpendicular to both a and b and satisfying d.c = 21.

Q.4. If a and b are unit vectors inclined at an angle θ then proved that: (i) cos θ/2 = ½|a + b| (ii) tan θ/2=|(a – b)/(a – b)|.

Q.5. If a, b, c are three mutually perpendicular vectors of equal magnitude. Prove that a + b + c is equally inclined with vectors a, b, and c. Also find angles.

Q.6. For any vector a prove that |a × i|² + |a × j|² + |a × k|² = 2|a|².

Q.7. Show that (a × b)² = |a|² |b|² – (a.b)².

Q.8. If a, b, and c are the position vectors of vertices A, B, C of a ΔABC, show that the area of a triangle ABC is ½|a × b + b × c + c × a|. Deduce the condition for points a, b, and c to be collinear.

Q.9. Let a, b and c be unit vectors such that a.b = a.c = 0 and the angle between b and c is π⁄6, prove that a = ±2(b × c ).

Q.10. If a, b and c are three vectors such that a + b + c = 0 , then prove that a × b = b × c = c × a.

Q.11. If a = i + j + k, c = j – k are given vectors, then find a vector b satisfying the equations a × b = c and a.b = 3.

Q.12. Let a, b and c be three non zero vectors such that c is a unit vector perpendicular to both a and b. if the angle between a and b is π⁄6, prove that [a b c]² = ¼|a|²|b|².

Q.13. If the vectors α = ai + j + k, β = i + bj + k and γ = i + j + ck are coplanar, than prove that 1/(1-a)+1/(1-b)+1/(1-c)=1 where a ≠ 1, b ≠ 1 and c ≠ 1.

Q.14. Find the altitude of a parallelepiped determined by the vectors a, b and c if the base taken as parallelogram determined by a and b and if a = i + j + k, b = 2i + 4j – k and c = i + j + 3k.

Q.15. Show that four points whose position vectors are 6i – 7j, 16i – 19j – 4k, 3i – 6k, 2i – 5j + 10k are coplanar.

Q.16. If |a| = 3, |b| = 4 and |c| = 5 such that each is perpendicular to sum of the other two, find |a + b + c|

Q.17. Decompose the vector 6i – 3j – 6k into vectors which are parallel and perpendicular to the vector i + j + k.

Q.18. If a, b and c are vectors such that a.b = a.c, a × b = a × c, a ≠ 0, then show that b = c.

Q.19. If a, b and c are three non zero vectors such that a × b = c and b × c = a. Prove that a, b and c are mutually at right angles and |b| = 1 and |c| = |a|.

Q.20. Simplify [a – b, b – c, c – a].

Q.21. If [abc] = 2, find the volume of the parallelepiped whose co-terminus edges are 2a + b, 2b + c, 2c + a.

Q.22. If a,b and c are three vectors such that a + b + c = 0 and |a| = 3, |b| = 5, |c| = 7, find the angle between a and b.

Q.23. The magnitude of the vector product of the vector i + j + k with a unit vector along the sum of the vector 2i + 4j + 5k and λi + 2j + 3k is equal to √2. Find the value of λ.

Q.24. If a × b = c × d and a × c = b × d, prove that (a – d) is parallel to (b – c), where a ≠ d and b ≠ c.

Q.25. Find a vector of magnitude √51 which makes equal angles with the vector a = 1/3 (i – (2j) + 2k), b = 1/5 (-4i – 3k) and c = j.

Q.26. If a,b and c are perpendicular to each other, then prove that [a b c] = a²b²c²

Q.27. If α = 3i – j and β = 3i + j + 3k then express β in the form of β = β1 + β2, where β1 is parallel to α and β2 is perpendicular to α.

Q.28. Find a unit vector perpendicular to plane ABC, when position vectors of A,B,C are (3i) – j + 2k, i – j – 3k and 4i – (3j) + k respectively.

Q.29. Find a unit vector in XY plane which makes an angle 45° with the vector i + j at angle 60° with the vector 3i – 4j.

Q.30. Find the altitude of a parallelepiped determined by the vectors a, b and c if the base taken as parallelogram determined by a and b and if a = i + j + k, b = 2i + 4j – k and c = i + j + 3k.

Q.31. Let v = (2i) + j – k and w = i + 3k. If u is a unit vector, then find the maximum value of the scalar triple product u, v, w.

Q.32. If a = i – k, b = xi + j + (1-x)k and c = yi + x j + (1 + x – y)k then prove that [a b c] depends upon neither x nor y.

Q.33. A, b and c are distinct non negative numbers, if the vectors ai + aj + ck and ci + cj + bk lie in a plane, then prove that c is the geometric mean of a and b.

Q.34. If |(a & a2 & 1+a3 @ b & b2 & 1+b3 @ c & c2 & 1+c3)| = 0 and vectors (1, a, a²), (1, b, b²) and (1, c, c²) are non-coplanar, then find the value of abc.

###### Previous Years CBSE Important Questions

1. Find the magnitude of each of the two vectors a and b, having the same magnitude such that the angle between them is 60º and their scalar product is 9/2. [CBSE Exam 2018]

2. If θ is the angle between two vectors i – 2j + 3k and 3i – 2j + k, find sin θ. [CBSE Exam 2018]

3. Let a = 4i + 5j – k , b = i – 4j + 5k and c = 3i + j – k. Find a vector d which is perpendicular to both a and b and satisfying d.c = 21.

4. What is the distance of the point (p, q, r) from the x-axis? [CBSE Sample Paper 2017]

5. 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. [Delhi 2017]

6. If a, b and c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b + c is equally inclined to a, b and c. Also, find the angle which a + b + c makes with a or b or c. [Delhi 2017]

7. Find the position vector of a point which divides the join of points with position vectors a – 2b and 2a + b externally in the ratio 2:1. [Delhi 2016]

8. The two vectors j + k and 3i – j + 4k represent the two sides AB and AC, respectively of a triangle ABC. Find the length of the median through A. [Delhi 2016]

9. Find a vector in the direction of a = i – 2j that has magnitude 7 units. [Delhi 2015C]

10. If a and b are unit vectors, then what is the angle between a and b so that √2 a – b is a unit vector? [Delhi 2015C]

11. If a = 7i + j – 4k and b = 2i + 6j + 3k, then find the projection of a on b. [Delhi 2015]

12. If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ. [Delhi 2015]

13. If r = xi + yj + zk, find (r × i).(r × j) + xy. [Delhi 2015]

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