NCERT solutions for class 12 Maths Chapter 10 exercise 10.1, 10.2, 10.3, 10.4 & miscellaneous exercises of Vector Algebra in PDF form for UP Board and CBSE Board students to free download. Download * Class 12 solutions all subjects*, Solutions are prepared according to

**CBSE Syllabus for 2018-2019**, sample papers along with board papers with solutions.

## NCERT solutions for class 12 Maths Chapter 10

**Click Here to NCERT Solutions Class 12 Maths**

### Class 12 Maths Solutions – Vector Algebra

#### NCERT Chapter to study online and answers given in the end of NCERT books.

#### These books are very good for revision and more practice. These book are also confined to NCERT Syllabus.

#### Assignments for practice

**Level 1 Test 1
**

**Level 2 Test 1
**

#### Important Questions Assignment with Solutions

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

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

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

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

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

Q.6. Write the value of p for which **a** = 3**i** + 2**j** + 9**k** and **b** = **i** + p**j** + 3**k** 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 (2**i** + 6**j** + 27**k**)×(**i** + 3**j** + p**k**) = **0**.

Q.9. Evaluate: **i**.(**j **× **k**) + (**i** × **k**).**j**.

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

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

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

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

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 3**i **+ **j** – 2**k** and **i** – 3**j** + 4**k**.

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 3**a** – 2**b** 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** = λ**i** + **j** + 4**k** on **b** = 2**i** + 6**j** + 3**k** is 4 units.

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

Q.21. If **a** = **i** – **j** + **k**, **b** = 2**i** + **j** – **k** and **c** = λ**i** – **j** + λ**k** 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 3**i** + 3**j** – 4**k** and 9**i** + 8**j** – 10**k** respectively?

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

##### Short Answers 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** = 2**i** – 2**j** + **k**, **b** = **i** + 2**j** – 2**k** and **c** = 2**i** – **j** + 4**k**.

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[**a**² + **b**²] then write the value of λ.

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

Q.9. Show that vectors **a** = 3**i** – 2**j** + **k**, **b** = **i** – 3**j** + 5**k**, **c** = 2**i** + **j** – 4**k** 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 3**i** – **j** + 4**k** 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 3**i** – y**j** + 2**k**, 5**i** – **j** + **k** and 3x**i** + 3**j** – **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** = 4**i** + 5**j** – **k** , **b** = **i** – 4**j** + 5**k** and **c** = 3**i** + **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** = a**i** + **j** + **k**, **β** = **i** + b**j** + **k** and **γ** = **i** + **j** + c**k** 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** = 2**i** + 4**j** – **k** and **c** = **i** + **j** + 3**k**.

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

**Download the Complete Solutions of the above Assignment HERE.**

#### Previous Years CBSE Important Questions

- 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] - If θ is the angle between two vectors
**i**– 2**j**+ 3**k**and 3**i**– 2**j**+**k**, find sin θ. [CBSE Exam 2018] - Let
**a**= 4**i**+ 5**j**–**k**,**b**=**i**– 4**j**+ 5**k**and**c**= 3**i**+**j**–**k**. Find a vector**d**which is perpendicular to both**a**and**b**and satisfying**d**.**c**= 21. - What is the distance of the point (p, q, r) from the x-axis? [CBSE Sample Paper 2017]
- 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]
- 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] - 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] - The two vectors
**j + k**and 3**i**–**j**+ 4**k**represent the two sides AB and AC, respectively of a triangle ABC. Find the length of the median through A. [Delhi 2016] - Find a vector in the direction of
**a**=**i**– 2**j**that has magnitude 7 units. [Delhi 2015C] - 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] - If
**a**= 7**i**+**j**– 4**k**and**b**= 2**i**+ 6**j**+ 3**k**, then find the projection of**a**on**b**. [Delhi 2015] - 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]
- If
**r**= x**i**+ y**j**+ z**k**, find (**r**×**i**).(**r**×**j**) + xy. [Delhi 2015]