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NCERT solutions for Class 12 Mathematics chapter 10 - Vector Algebra

Chapter 10 - Vector Algebra

Page 428

Q 1 | Page 428

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

Q 2 | Page 428

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

Q 3 | Page 428

Classify the following as scalar and vector quantities.

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v) work done

Q 4 | Page 428

In Figure, identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

Q 5.1 | Page 428

Answer the following as true or false.

Two vectors having same magnitude are collinear.

Q 5.2 | Page 428

Answer the following as true or false.

veca and -veca are collinear

Q 5.2 | Page 428

Answer the following as true or false.

Two collinear vectors are always equal in magnitude.

Q 5.4 | Page 428

Answer the following as true or false.

Two collinear vectors having the same magnitude are equal

Pages 440 - 441

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

Q 2 | Page 440

Write two different vectors having same magnitude.

Q 3 | Page 440

Write two different vectors having same direction.

Q 4 | Page 440

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

Q 5 | Page 440

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

Q 6 | Page 440

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

Q 7 | Page 440

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

Q 8 | Page 440

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

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

Q 10 | Page 440

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

Q 11 | Page 440

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

Q 12 | Page 440

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

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.

Q 14 | Page 440

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

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 ration 2:1

(i) internally

(ii) externally

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).

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.

Q 18 | Page 441

In triangle ABC which of the following is not true:

(A) AB + BC + CA = 0

(B) AB+ BC −AC = 0

(C) AB+ BC −CA = 0

(D) AB−CB+CA = 0

Q 19 | Page 441

If a and b are two collinear vectors, then which of the following are incorrect:

(A) b = λa, for some scalar λ

(B) a = ±b

(C) the respective components of a and b are not proportional

(D) both the vectors a and b have same direction, but different magnitudes.

Pages 447 - 448

Q 1 | Page 447

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

Q 2 | Page 447

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

Q 3 | Page 447

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

Q 4 | Page 447

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

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.

Q 6 | Page 448

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

Q 7 | Page 448

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

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

Q 9 | Page 448

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

Q 10 | Page 448

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

Q 11 | Page 448

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

Q 12 | Page 448

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

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

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.

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)

Q 16 | Page 448

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

Q 17 | Page 448

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

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|

Pages 454 - 455

Q 1 | Page 454

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

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.

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

Q 4 | Page 454

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

Q 5 | Page 454

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

Q 6 | Page 454

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

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

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.

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).

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

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

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

Pages 458 - 459

Q 1 | Page 458

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

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)

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.

Q 4 | Page 458

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

Q 5 | Page 458

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

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

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

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.

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 (2veca + vecb) and (veca - 3vecb) externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

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.

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

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.

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.

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.

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

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

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

(A) theta = pi/4

(B) theta = pi/3

(C) theta =pi/2

(D) theta = 2pi/3

Q 18 | Page 459

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

(A) 0

(B) –1

(C) 1

(D) 3

Q 19 | Page 459

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

(A) 0

(B) pi/4

(C) pi/4`

(D) π

NCERT solutions for Class 12 Mathematics chapter 10 - Vector Algebra

NCERT solutions for Class 12 Mathematics chapter 10 (Vector Algebra) include all questions with solution and detail explanation from Mathematics Textbook for Class 12 Part 2. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics Textbook for Class 12 Part 2 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. These NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 10 Vector Algebra are Addition of Vectors, Multiplication of a Vector by a Scalar, Concept of 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, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Types of Vectors, Components of a 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 solutions for Class 12 Mathematics by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

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