NCERT solutions for Class 12 Maths chapter 10 - Vector Algebra [Latest edition]

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Solutions for Chapter 10: Vector Algebra

Below listed, you can find solutions for Chapter 10 of CBSE, Karnataka Board PUC NCERT for Class 12 Maths.

Exercise 10.1Exercise 10.2Exercise 10.3Exercise 10.4Exercise 10.5
Exercise 10.1 [Page 428]

NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.1 [Page 428]

Exercise 10.1 | Q 1 | Page 428

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

Exercise 10.1 | Q 2. (i) | Page 428

Classify the following measures as scalar and vector.

10 kg

Exercise 10.1 | Q 2. (ii) | Page 428

Classify the following measures as scalar and vector.

2 meters north-west

Exercise 10.1 | Q 2. (ii) | Page 428

Classify the following measures as scalar and vector.

40°

Exercise 10.1 | Q 2. (iv) | Page 428

Classify the following measures as scalar and vector.

40 watt

Exercise 10.1 | Q 2. (v) | Page 428

Classify the following measures as scalar and vector.

10-19 coulomb

Exercise 10.1 | Q 2. (vi) | Page 428

Classify the following measures as scalar and vector.

20 m/s2

Exercise 10.1 | Q 3. (i) | Page 428

Classify the following as scalar and vector quantity.

Time period

Exercise 10.1 | Q 3. (ii) | Page 428

Classify the following as scalar and vector quantity.

distance

Exercise 10.1 | Q 3. (iii) | Page 428

Classify the following as scalar and vector quantity.

Force

Exercise 10.1 | Q 3. (iv) | Page 428

Classify the following as scalar and vector quantity.

velocity

Exercise 10.1 | Q 3. (v) | Page 428

Classify the following as scalar and vector quantity.

work done

Exercise 10.1 | Q 4. (i) | Page 428

In Figure, identify the following vector.

Coinitial

Exercise 10.1 | Q 4. (ii) | Page 428

In Figure, identify the following vector.

Equal

Exercise 10.1 | Q 4. (iii) | Page 428

In Figure, identify the following vector.

Collinear but not equal

Answer the following as true or false.

Exercise 10.1 | Q 5. (i) | Page 428

veca and -veca are collinear.

• True

• False

Exercise 10.1 | Q 5. (ii) | Page 428

Two collinear vectors are always equal in magnitude.

• True

• False

Exercise 10.1 | Q 5. (iii) | Page 428

Two vectors having the same magnitude are collinear.

• True

• False

Exercise 10.1 | Q 5. (iv) | Page 428

Two collinear vectors having the same magnitude are equal.

• True

• False

Exercise 10.2 [Pages 440 - 441]

NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.2 [Pages 440 - 441]

Exercise 10.2 | Q 1. | Page 440

Compute the magnitude of the following vectors:

veca = hati + hatj + hatk; vecb = 2hati - 7hatj - 3hatk;  vecc = 1/sqrt3 hati + 1/sqrt3 hatj - 1/sqrt3 hatk

Exercise 10.2 | Q 2. | Page 440

Write two different vectors having same magnitude.

Exercise 10.2 | Q 3. | Page 440

Write two different vectors having same direction.

Exercise 10.2 | Q 4. | Page 440

Find the values of x and y so that the vectors 2hati + 3hatj and xhati  + yhatj are equal.

Exercise 10.2 | Q 5. | Page 440

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

Exercise 10.2 | Q 6. | Page 440

Find the sum of the vectors veca = hati -2hatj + hatk, vecb = -2hati + 4hatj + 5hatk and vecc = hati - 6hatj - 7hatk

Exercise 10.2 | Q 7. | Page 440

Find the unit vector in the direction of the vector veca = hati + hatj + 2hatk.

Exercise 10.2 | Q 8. | Page 440

Find the unit vector in the direction of vector vec(PQ), where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Exercise 10.2 | Q 9. | Page 440

For given vectors,  veca = 2hati - hatj + 2hatk and vecb = -hati  + hatj - hatk, find the unit vector in the direction of the vector veca +vecb.

Exercise 10.2 | Q 10. | Page 440

Find a vector in the direction of vector 5hati - hatj +2hatk which has a magnitude of 8 units.

Exercise 10.2 | Q 11. | Page 440

Show that the vectors 2hati - 3hatj + 4hatk and -4hati + 6hatj -  8hatk are collinear.

Exercise 10.2 | Q 12. | Page 440

Find the direction cosines of the vector hati + 2hatj + 3hatk.

Exercise 10.2 | Q 13. | Page 440

Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.

Exercise 10.2 | Q 14. | Page 440

Show that the vector hati + hatj + hatk is equally inclined to the axes OX, OY, and OZ.

Exercise 10.2 | Q 15. | Page 440

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  hati + 2hatj - hatk and -hati + hatj + hatk  respectively, in the ratio 2:1

(i) internally

(ii) externally

Exercise 10.2 | Q 16. | Page 441

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

Exercise 10.2 | Q 17. | Page 441

Show that the points A, B and C with position vectors veca = 3hati - 4hatj - 4hatk, vecb = 2hati - hatj + hatk and vecc = hati - 3hatj - 5hatk, respectively form the vertices of a right angled triangle.

Exercise 10.2 | Q 18. | Page 441

In triangle ABC which of the following is not true:

(A) vec(AB) + vec(BC) + vec(CA) = vec0

(B) vec(AB) + vec(BC) - vec(AC) = vec0

(C) vec(AB) + vec(BC) - vec(AC) = 0

(D) vec(AB) - vec(CB) + vec(CA) = vec0

Exercise 10.2 | Q 19. | Page 441

If veca and vecb are two collinear vectors, then which of the following are incorrect:

(A) vecb = λveca, for some scalar λ

(B) veca = ±vecb

(C) the respective components of veca and vecb are not proportional.

(D) both the vectors veca and vecb have the same direction, but different magnitudes.

Exercise 10.3 [Pages 447 - 448]

NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 [Pages 447 - 448]

Exercise 10.3 | Q 1 | Page 447

Find the angle between two vectors veca and vecb with magnitudes sqrt3 and 2, respectively having veca.vecb = sqrt6.

Exercise 10.3 | Q 2 | Page 447

Find the angle between the vectors hati - 2hatj + 3hatk and 3hati - 2hatj + hatk

Exercise 10.3 | Q 3 | Page 447

Find the projection of the vector hati - hatj on the vector hati + hatj.

Exercise 10.3 | Q 4 | Page 447

Find the projection of the hati + 3hatj + 7hatk  on the vector 7hati - hatj + 8hatk

Exercise 10.3 | Q 5 | Page 447

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

1/7 (2hati + 3hatj + 6hatj), 1/7(3hati - 6hatj + 2hatk), 1/7(6hati + 2hatj - 3hatk)

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

Exercise 10.3 | Q 6 | Page 448

Find |veca| and |vecb|, if (veca + vecb).(veca -vecb) = 8 and |veca| = 8|vecb|

Exercise 10.3 | Q 7 | Page 448

Evaluate the product (3veca - 5vecb).(2veca + 7vecb)

Exercise 10.3 | Q 8 | Page 448

Find the magnitude of two vectors veca and vecb , having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2

Exercise 10.3 | Q 9 | Page 448

Find |vecx|, if for a unit vector veca , (vecx -  veca).(vecx + veca) = 12

Exercise 10.3 | Q 10 | Page 448

If veca = 2hati + 2hatj + 3hatk , vecb = -veci + 2hatj + hatk and vecc = 3hati + hatj are such that the vector(veca + lambdavecb) is perpendicular to vecc, then find the value of λ.

Exercise 10.3 | Q 11 | Page 448

Show that |veca|vecb+|vecb| is perpendicular to |veca|vecb-|vecb|veca, for any two nonzero vectors veca and vecb

Exercise 10.3 | Q 12 | Page 448

If  veca.veca = 0 and veca.vecb = 0 , then what can be concluded about the vector vecb?

Exercise 10.3 | Q 13 | Page 448

If veca","vecb","veccare unit vectors such that veca+vecb+vecc=0, then write the value of  veca.vecb+vecb.vecc+vecc.veca

Exercise 10.3 | Q 14 | Page 448

If either vector veca = vec0  or vecb = vec0 , then veca.vecb = 0. But the converse need not be true. Justify your answer with an example.

Exercise 10.3 | Q 15 | Page 448

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 vectors bar(BA) and bar(BC)

Exercise 10.3 | Q 16 | Page 448

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

Exercise 10.3 | Q 17 | Page 448

Show that the vectors 2hati - hatj + hatk and 3hati - 4hatj - 4hatk form the vertices of a right-angled triangle.

Exercise 10.3 | Q 18 | Page 448

If veca is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λveca is unit vector if

(A) λ = 1 (B) λ = –1

(C) a = |λ|

(D) a = 1/|λ|

Exercise 10.4 [Pages 454 - 455]

NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.4 [Pages 454 - 455]

Exercise 10.4 | Q 1 | Page 454

Find |a ×b|, if veca = hati - 7hatj + 7hatk and vecb = 3hati - 2hatj + 2hatk
.

Exercise 10.4 | Q 2 | Page 454

Find a unit vector perpendicular to each of the vector  veca  + vecb and veca - vecb, where veca = 3hati + 2hatj + 2hatk and vecb = hati + 2hatj  - 2hatk.

Exercise 10.4 | Q 3 | Page 454

If a unit vector veca makes an angles pi/3 with hati, pi/4 with hatj and an acute angle θ with hatk, then find θ and hence, the compounds of veca

Exercise 10.4 | Q 4 | Page 454

Show that (veca - vecb) xx (veca + vecb) = 2(veca xx vecb)

Exercise 10.4 | Q 5 | Page 454

Find λ and μ if  (2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0

Exercise 10.4 | Q 6 | Page 454

Given that veca.vecb = 0 and veca xx vecb = 0 What can you conclude about the vectors veca and vecb?

Exercise 10.4 | Q 7 | Page 454

Let the vectors veca, vecb, vecc given as a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk Then show that = veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc

Exercise 10.4 | Q 8 | Page 454

If either veca = vec0  or vecb = vec0, then veca xxvecb = vec0. Is the converse true? Justify your answer with an example.

Exercise 10.4 | Q 9 | Page 454

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

Exercise 10.4 | Q 10 | Page 455

Find the area of the parallelogram whose adjacent sides are determined by the vector veca = hati - hatj + 3hatk and vecb = 2hati - 7hatj + hatk

Exercise 10.4 | Q 11 | Page 455

Let the vectors veca and vecb be such that |veca| = 3 and |vecb| = sqrt2/3 , then veca xx vecb is a unit vector, if the angle between veca and vecb is

(A) pi/6

(B) pi/4

(C) pi/3

(D) pi/2

Exercise 10.4 | Q 12 | Page 455

Area of a rectangle having vertices A, B, C, and D with position vectors -hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk and  respectively is

(A) 1/2

(B) 1

(C)2

(D) 4

Exercise 10.5 [Pages 458 - 459]

NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.5 [Pages 458 - 459]

Exercise 10.5 | Q 1 | Page 458

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

Exercise 10.5 | Q 2 | Page 458

Find the scalar components and magnitude of the vector joining the points P(x_1, y_1, z_1) and Q (x_2, y_2 , z_2)

Exercise 10.5 | Q 3 | Page 458

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.

Exercise 10.5 | Q 4 | Page 458

If veca = vecb + vecc, then is it true that |veca| = |vecb| + |vecc|? Justify your answer.

Exercise 10.5 | Q 5 | Page 458

Find the value of x for which x(hati + hatj + hatk) is a unit vector.

Exercise 10.5 | Q 6 | Page 458

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors veca = 2i + 3hatj - hatk and vecb = hati - 2hatj + hatk

Exercise 10.5 | Q 7 | Page 458

if veca = hati  +hatj + hatk, vecb = 2hati - hatj +  3hatk and vecc = hati - 2hatj + hatk find a unit vector parallel to the vector 2veca - vecb + 3vecc

Exercise 10.5 | Q 8 | Page 458

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.

Exercise 10.5 | Q 9 | Page 458

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are P(2veca + vecb) and Q(veca - 3vecb) externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Exercise 10.5 | Q 10 | Page 458

The two adjacent sides of a parallelogram are 2hati - 4hatj + 5hatk and hati - 2hatj - 3hatk. Find the unit vector parallel to its diagonal. Also, find its area.

Exercise 10.5 | Q 11 | Page 458

Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are 1/sqrt3, 1/sqrt3, 1/sqrt3.

Exercise 10.5 | Q 12 | Page 458

Let veca = hati + 4hatj + 2hatk, vecb = 3hati - 2hatj + 7hatk  and vecc = 2hati - hatj + 4hatk. Find a vector vecd which is perpendicular to both veca and vecb, and vecc.vecd = 15.

Exercise 10.5 | Q 13 | Page 458

The scalar product of the vector hati + hatj + hatk with a unit vector along the sum of vectors 2hati + 4hatj - 5hatk and  lambdahati + 2hatj +  3hatk is equal to one. Find the value of lambda.

Exercise 10.5 | Q 14 | Page 458

If veca, vecb, vecc are mutually perpendicular vectors of equal magnitudes, show that the vector veca +  vecb+ vecc is equally inclined to veca, vecb and vecc.

Exercise 10.5 | Q 15 | Page 459

Prove that (veca + vecb).(veca + vecb) = |veca|^2 + |vecb|^2 if and only if veca.vecb are perpendicular, given veca != vec0, vecb != vec0

Exercise 10.5 | Q 16 | Page 459

If θ is the angle between two vectors veca and vecb, then veca.vecb >= 0 only when

(A) 0 < theta < pi/2

(B) 0 <= theta <= pi/2

(C) 0 < theta < pi

(D) 0 <= theta <= pi

Exercise 10.5 | Q 17 | Page 459

Let veca and vecb be two unit vectors andθ is the angle between them. Then veca + vecb is a unit vector if ______

• theta = pi/4

• theta = pi/3

• theta =pi/2

• theta = (2pi)/3

Exercise 10.5 | Q 18 | Page 459

The value of is hati.(hatj xx hatk)+hatj.(hatixxhatk)+hatk.(hatixxhatj).

• 0

• -1

• 1

• 3

Exercise 10.5 | Q 19 | Page 459

If θ is the angle between any two vectors veca and vecb , then |veca.vecb| = |veca xx vecb| when θ is equal to ______

• 0

• pi/4

• pi/2`

• π

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Solutions for Chapter 10: Vector Algebra

Exercise 10.1Exercise 10.2Exercise 10.3Exercise 10.4Exercise 10.5

NCERT solutions for Class 12 Maths chapter 10 - Vector Algebra

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 12 Maths CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC 10 (Vector Algebra) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Class 12 Maths chapter 10 Vector Algebra are Direction Cosines, Properties of Vector Addition, Geometrical Interpretation of Scalar, Scalar Triple Product of Vectors, Vector (Or Cross) Product of Two Vectors, Scalar (Or Dot) Product of Two Vectors, Position Vector of a Point Dividing a Line Segment in a Given Ratio, Multiplication of a Vector by a Scalar, Addition of Vectors, Vectors and Their Types, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Components of Vector, Section Formula, Vector Joining Two Points, Vectors Examples and Solutions, Projection of a Vector on a Line, Introduction of Product of Two Vectors.

Using NCERT Class 12 Maths solutions Vector Algebra exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Class 12 Maths students prefer NCERT Textbook Solutions to score more in exams.

Get the free view of Chapter 10, Vector Algebra Class 12 Maths additional questions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.

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