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NCERT solutions Mathematics Class 12 Part 2 chapter 10 Vector Algebra

Chapters

NCERT Solutions for Mathematics Class 12 Part 1

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2

Chapter 10 - Vector Algebra

Page 428

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

Q 1 | 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 2 | Page 428

Classify the following as scalar and vector quantities.

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v) work done

Q 3 | Page 428

In Figure, identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

Q 4 | Page 428

Answer the following as true or false.

Two vectors having same magnitude are collinear.

Q 5.1 | Page 428

Answer the following as true or false.

Two collinear vectors are always equal in magnitude.

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 having the same magnitude are equal

Q 5.4 | Page 428

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

Write two different vectors having same magnitude.

Q 2 | Page 440

Write two different vectors having same direction.

Q 3 | Page 440

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

Q 4 | Page 440

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

Q 5 | Page 440

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

Q 6 | Page 440

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

Q 7 | 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 8 | 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 9 | Page 440

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

 
Q 10 | Page 440

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

Q 11 | Page 440

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

Q 12 | 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 13 | Page 440

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

 
Q 14 | 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 15 | Page 440

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

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

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

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

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

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

Q 2 | Page 447

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

Q 3 | Page 447

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

Q 4 | 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 5 | Page 447

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

Q 6 | Page 448

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

Q 7 | 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 8 | Page 448

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

Q 9 | 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 10 | Page 448

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

 
Q 11 | Page 448

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

Q 12 | Page 448

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

Q 13 | 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 14 | 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 15 | Page 448

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

Q 16 | Page 448

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

 
Q 17 | Page 448

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

(A) λ = 1 (B) λ = –1

(C) a = |λ|

(D) a = 1/|λ|

Q 18 | Page 448

Pages 454 - 455

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

Q 1 | 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 2 | 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 3 | Page 454

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

Q 4 | Page 454

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

Q 5 | Page 454

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

Q 6 | 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 7 | Page 454

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

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

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

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

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

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

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

 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

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

 
Q 4 | Page 458

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

Q 5 | 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 6 | 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 7 | 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 8 | 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 9 | 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 10 | 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 11 | 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 12 | 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 13 | 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 14 | Page 458

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

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

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

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

(A) 0

(B) –1

(C) 1

(D) 3

Q 18 | 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

Q 19 | Page 459

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2
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