NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

Class XII Mathematics Chapter 10 exercise 10.1, 10.2, 10.3, 10.4 and miscellaneous exercises in Hindi and English Medium for UP Board and CBSE Board students. Download Class 12 solutions all subjects, Solutions are prepared according to CBSE Syllabus for 2024-25. Solutions are prepared as per the CBSE guidelines and are easy to understand. Videos of solutions in Hindi and English are also given here to explore each exercise.

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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.

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.

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.

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]