NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra

The NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra are given here where the students learn about the difference between a scalar and a vector quantity, their properties, operations of vectors etc. The topic has an important role in helping the students score high marks not only in the board exams but also the competitive exams.

It is important to be prepared for the various problems asked during the 12th board examination. Solving through different exercises and problem sets give students the confidence to write the exams better. These class 12 NCERT solutions for Vector Algebra are very easy to understand. Students can also avail these NCERT solutions and download it for free to practice them offline as well.

Download PDF of NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra

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NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

Exercise 10.1 Page No: 428

1. Represent graphically a displacement of 40 km, 30° east of north.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 1

The vector
NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 2represents the displacement of 40 km, 30o east of north.

2. Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2

Solution:

(i) 10 kg is a scalar quantity because it has only magnitude.

(ii) 2 meters north-west is a vector quantity as it has both magnitude and direction.

(iii) 40° is a scalar quantity as it has only magnitude.

(iv) 40 watts is a scalar quantity as it has only magnitude.

(v) 10–19 coulomb is a scalar quantity as it has only magnitude.

(vi) 20 m/s2 is a vector quantity as it has both magnitude and direction.

3. Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

Solution:

(i) Time period is a scalar quantity as it has only magnitude.

(ii) Distance is a scalar quantity as it has only magnitude.

(iii) Force is a vector quantity as it has both magnitude and direction.

(iv) Velocity is a vector quantity as it has both magnitude as well as direction.

(v) Work done is a scalar quantity as it has only magnitude.

4. In Figure, identify the following vectors.

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 3

(i) Coinitial (ii) Equal (iii) Collinear but not equal

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 4directions are not the same.

5. Answer the following as true or false.

(i) NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 5 andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 6are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Solution:

(i) True.

Vectors 
NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 7 and
NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.1 - 8are parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

Two vectors having the same magnitude need not necessarily be parallel to the same line.

(iv) False.

Only if the magnitude and direction of two vectors are the same, regardless of the positions of their initial points the two vector are said to be equal.


Exercise 10.2 Page No: 440

1. Compute the magnitude of the following vectors:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 1

Solution:

Given vectors are:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 2

2. Write two different vectors having same magnitude.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 3

3. Write two different vectors having same direction.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 4

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 5

4. Find the values of x and y so that the vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 6are equal

Solution:

Given vectors 
NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 7will be equal only if their corresponding components are equal.

Thus, the required values of x and y are 2 and 3 respectively.

5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Solution:

The vector with initial point P (2, 1) and terminal point Q (–5, 7) can be shown as,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 8

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 9Thus, the required scalar components are –7 and 6 while the vector components are 

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 106. Find the sum of the vectors

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 11

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 127. Find the unit vector in the direction of the vector

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 13

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 14

8. Find the unit vector in the direction of vector , where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 15

9. For given vectors, NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 16and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 17, find the unit vector in the direction of the vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 18

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 19

10. Find a vector in the direction of vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 20which has magnitude 8 units.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 21

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 22

11. Show that the vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 23are collinear.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 24

Therefore, the given vectors are collinear.

12. Find the direction cosines of the vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 25

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 26

13. Find the direction cosines of the vector joining the points A (1, 2, –3) and

B (–1, –2, 1) directed from A to B.

Solution:

Given points are A (1, 2, –3) and B (–1, –2, 1).

Now,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 27

14. Show that the vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 28is equally inclined to the axes OX, OY, and OZ.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 29

15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 30 respectively, in the ration 2:1

(i) internally

(ii) externally

Solution:

The position vector of point R dividing the line segment joining two points

P and Q in the ratio m: is given by:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 31

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 32

16. Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

Solution:

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 33

17. Show that the points A, B and C with position vectors,NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 34NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 35respectively form the vertices of a right angled triangle.

Solution:

Given position vectors of points A, B, and C are:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 36

18. In triangle ABC (Fig 10.18) which of the following is not true:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 37

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 38

19. If NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 39are two collinear vectors, then which of the following are incorrect:

A. NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 40, for some scalar λ

B. NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 41

C. the respective components of NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 42are proportional

D. both the vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 43have same direction, but different magnitudes

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 44

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.2 - 45


Exercise 10.3 Page No: 447

1. Find the angle between two vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 1andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 2with magnitudes √3 and 2, respectively havingNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 3.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 4

2. Find the angle between the vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 5

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 6

3. Find the projection of the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 7on the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 8.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 9

4. Find the projection of the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 10on the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 11.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 12

5. Show that each of the given three vectors is a unit vector:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 13

Also, show that they are mutually perpendicular to each other.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 14

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 15

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 166. Find

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 17

7. Evaluate the productNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 18

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 19

8. Find the magnitude of two vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 20, having the same magnitude and such that the angle between them is 60° and their scalar product is ½.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 21

9. FindNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 23, if for a unit vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 22

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 24

10. IfNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 25are such thatNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 26is perpendicular toNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 27, then find the value of λ.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 28

11. Show that NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 29is perpendicular toNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 30, for any two nonzero vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 31.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 32

12. IfNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 33, then what can be concluded about the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 34?

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 35

13. If NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 36are unit vectors such that NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 37, find the value of NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 38.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 39

14. If either vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 40, thenNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 41. But the converse need not be true. Justify your answer with an example.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 42

15. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 43andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 44]

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 45

16. Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Solution:

Given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

Now,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 46

Therefore, the given points A, B, and C are collinear.

17. Show that the vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 47form the vertices of a right angled triangle.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 48

18. IfNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra - 2is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 49is unit vector if

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 50

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.3 - 51


Exercise 10.4 Page No: 454

1. FindNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 1, if NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 2andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 3

Solution:

We have,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 4

2. Find a unit vector perpendicular to each of the vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 5andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 6, where NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 7andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 8.

Solution:

We have,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 9

3. If a unit vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 10 makes an anglesNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 11with NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 12with NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 13and an acute angle θ withNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 14, then find θ and hence, the compounds ofNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra - 1.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 15

4. Show that

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 16

Solution:

Taking the LHS, we have

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 17

5. Find λ and μ if NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 18.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 19

6. Given that NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 20 andif. What can you conclude about the vectorsNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 22?

Solution:

Given,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 23

7. Let the vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 24given asNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 25 NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 26. Then show that NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 27

Solution:

We have,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 28

8. If either NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 29 orNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 30, thenNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 31. Is the converse true? Justify your answer with an example.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 32

9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 33

10. Find the area of the parallelogram whose adjacent sides are determined by the vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 34.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 35

11. Let the vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 36and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 37be such that NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 38andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 39, thenNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 40 is a unit vector, if the angle between NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 41and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 42is

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 43

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 44

12. Area of a rectangle having vertices A, B, C, and D with position vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 45and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 46 respectively is

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 47

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra ex.10.4 - 48


Miscellaneous Exercise Page No: 458

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 1

2. Find the scalar components and magnitude of the vector joining the points P (x1, y1, z1) and Q (x2, y2, z2).

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 2

3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Solution:

Let O and B be the initial and final positions of the girl respectively.

Then, the girl’s position can be shown as:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 3

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 4

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 94. IfNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 5, then is it true thatNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 6? Justify your answer.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 7

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 8

5. Find the value of x for whichNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 10is a unit vector.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 11

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 12.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 13

7. IfNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 14, find a unit vector parallel to the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 15.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 16

8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 17

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 18

9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors areNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 19externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 20

10. The two adjacent sides of a parallelogram areNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 21and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 22.

Find the unit vector parallel to its diagonal. Also, find its area.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 23

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 24

11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ areNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 25.

Solution:

Let’s assume a vector to be equally inclined to axes OX, OY, and OZ at angle α.

Then, the direction cosines of the vector are cos α, cos α, and cos α.

Now, we know that

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 26

Therefore, the direction cosines of the vector which are equally inclined to the axes are

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 27.

12. Let NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 28andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 29. Find a vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 30which is perpendicular to both NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 31andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 32, andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 33.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 34

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 35

13. The scalar product of the vectorNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 36with a unit vector along the sum of vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 37and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 38is equal to one. Find the value of λ.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 39

14. If NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 40are mutually perpendicular vectors of equal magnitudes, show that the vector NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 41is equally inclined to NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 42andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 43.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 44

15. Prove thatNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 45, if and only if NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 46are perpendicular, givenNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 47.

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 48

16. If θ is the angle between two vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 49and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 50, then NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 51only when

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 52

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 55

17. Let NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 56 and NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 57 be two unit vectors andθ is the angle between them. Then NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 58is a unit vector if

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 59

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 60

18. The value of NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 61is

(A) 0 (B) –1 (C) 1 (D) 3

Solution:

Given,

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 62

The correct answer is C.

19. If θ is the angle between any two vectors NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 63andNCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 64, then NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 65when θ is equal to

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 66

Solution:

NCERT Solutions Class 11 Mathematics Chapter 10 Vector Algebra miscellaneous ex- 67

The major concepts of Maths covered in Chapter 10- Vector Algebra of NCERT Solutions for Class 12 includes:

10.1 Introduction

10.2 Basic Concepts

    • Position Vector
    • Direction Cosines

10.3 Types of Vectors

    • Zero Vector
    • Unit Vector
    • Coinitial Vectors
    • Collinear Vectors
    • Equal Vectors
    • Negative of a Vector

10.4 Addition of Vectors

    • Properties of vector addition

10.5 Multiplication of a Vector by a Scalar

10.5.1 Components of a vector

10.5.2 Vector joining two points

10.5.3 Section formula

10.6 Product of Two Vectors

10.6.1 Scalar (or dot) product of two vectors

10.6.2 Projection of a vector on a line

10.6.3 Vector (or cross) product of two vectors

Exercise 10.1 Solutions 5 Questions

Exercise 10.2 Solutions 19 Questions

Exercise 10.3 Solutions 18 Questions

Exercise 10.4 Solutions 12 Questions

Miscellaneous Exercise On Chapter 10 Solutions 19 Questions

NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra

The chapter Vector Algebra belongs to the unit Vectors and Three – Dimensional Geometry, that adds up to 14 marks of the total marks. There are 4 exercises along with a miscellaneous exercise in this chapter to help students understand the concepts related to Vectors and Vector Algebra clearly. Some of the topics discussed in the tenth Chapter of NCERT Solutions for Class 12 Maths are as follows:

  1. The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.
  2. The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as l=(a/r), m=(b/r) n=(c/r)
  3. The vector sum of the three sides of a triangle taken in order is 0.
  4. The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
  5. The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) according as the value of λ is positive (or negative).

The other concepts and topics explained in the chapter can be understood by going through the Chapter 10 of the NCERT textbook for class 12.

Key Features of NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra

Studying the Vector Algebra of Class 12 enables the students to understand the following:

Vectors and scalars, magnitude and direction of a vector.Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

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