# NCERT solutions Mathematics Textbook for Class 12 Part 2 chapter 4 Vector Algebra

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 75- 180-

### Chapter 4 - Vector Algebra

#### Page 428

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

Q 1 | Page 428 | view solution

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 2 | Page 428 | view solution

Classify the following as scalar and vector quantities.

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v) work done

Q 3 | Page 428 | view solution

In Figure, identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

Q 4 | Page 428 | view solution

Answer the following as true or false.

Two vectors having same magnitude are collinear.

Q 5.1 | Page 428 | view solution

Answer the following as true or false.

Two collinear vectors are always equal in magnitude.

Q 5.2 | Page 428 | view solution

Answer the following as true or false.

veca and -veca are collinear

Q 5.2 | Page 428 | view solution

Answer the following as true or false.

Two collinear vectors having the same magnitude are equal

Q 5.4 | Page 428 | view solution

#### Pages 440 - 441

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 1 | Page 440 | view solution

Write two different vectors having same magnitude.

Q 2 | Page 440 | view solution

Write two different vectors having same direction.

Q 3 | Page 440 | view solution

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

Q 4 | Page 440 | view solution

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

Q 5 | Page 440 | view solution

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

Q 6 | Page 440 | view solution

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

Q 7 | Page 440 | view solution

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 8 | Page 440 | view solution

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 9 | Page 440 | view solution

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

Q 10 | Page 440 | view solution

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

Q 11 | Page 440 | view solution

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

Q 12 | Page 440 | view solution

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 13 | Page 440 | view solution

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

Q 14 | Page 440 | view solution

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 15 | Page 440 | view solution

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

Q 16 | Page 441 | view solution

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 17 | Page 441 | view solution

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 18 | Page 441 | view solution

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.

Q 19 | Page 441 | view solution

#### Pages 447 - 448

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

Q 1 | Page 447 | view solution

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

Q 2 | Page 447 | view solution

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

Q 3 | Page 447 | view solution

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

Q 4 | Page 447 | view solution

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 5 | Page 447 | view solution

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

Q 6 | Page 448 | view solution

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

Q 7 | Page 448 | view solution

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 8 | Page 448 | view solution

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

Q 9 | Page 448 | view solution

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 10 | Page 448 | view solution

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

Q 11 | Page 448 | view solution

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

Q 12 | Page 448 | view solution

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

Q 13 | Page 448 | view solution

If either vector veca = vec0or vecb = vec0 , then veca.vecb = 0. But the converse need not be true. Justify your answer with an example.

Q 14 | Page 448 | view solution

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 15 | Page 448 | view solution

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

Q 16 | Page 448 | view solution

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

Q 17 | Page 448 | view solution

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|

Q 18 | Page 448 | view solution

#### Pages 454 - 455

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

Q 1 | Page 454 | view solution

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 2 | Page 454 | view solution

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 3 | Page 454 | view solution

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

Q 4 | Page 454 | view solution

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

Q 5 | Page 454 | view solution

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

Q 6 | Page 454 | view solution

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 7 | Page 454 | view solution

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

Q 8 | Page 454 | view solution

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

Q 9 | Page 454 | view solution

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

Q 10 | Page 455 | view solution

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 11 | Page 455 | view solution

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

Q 12 | Page 455 | view solution

#### Pages 458 - 459

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

Q 1 | Page 458 | view solution

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 2 | Page 458 | view solution

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 3 | Page 458 | view solution

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

Q 4 | Page 458 | view solution

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

Q 5 | Page 458 | view solution

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 6 | Page 458 | view solution

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 7 | Page 458 | view solution

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 8 | Page 458 | view solution

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 9 | Page 458 | view solution

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 10 | Page 458 | view solution

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 11 | Page 458 | view solution

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 12 | Page 458 | view solution

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 13 | Page 458 | view solution

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 14 | Page 458 | view solution

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

Q 15 | Page 459 | view solution

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 16 | Page 459 | view solution

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 17 | Page 459 | view solution

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

(A) 0

(B) –1

(C) 1

(D) 3

Q 18 | Page 459 | view solution

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

Q 19 | Page 459 | view solution