#### Online Mock Tests

#### Chapters

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity and Differentiability

Chapter 6: Application of Derivatives

Chapter 7: Integrals

Chapter 8: Application of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

## 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.

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

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

Classify the following measures as scalar and vector.

10 kg

Classify the following measures as scalar and vector.

2 meters north-west

Classify the following measures as scalar and vector.

40°

Classify the following measures as scalar and vector.

40 watt

Classify the following measures as scalar and vector.

10^{-19} coulomb

Classify the following measures as scalar and vector.

20 m/s^{2}

Classify the following as scalar and vector quantity.

Time period

Classify the following as scalar and vector quantity.

distance

Classify the following as scalar and vector quantity.

Force

Classify the following as scalar and vector quantity.

velocity

Classify the following as scalar and vector quantity.

work done

In Figure, identify the following vector.

Coinitial

In Figure, identify the following vector.

Equal

In Figure, identify the following vector.

Collinear but not equal

#### Answer the following as true or false.

`veca and -veca` are collinear.

True

False

Two collinear vectors are always equal in magnitude.

True

False

Two vectors having the same magnitude are collinear.

True

False

Two collinear vectors having the same magnitude are equal.

True

False

### NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.2 [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`

Write two different vectors having same magnitude.

Write two different vectors having same direction.

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

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

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

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

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.

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

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

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

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

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

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

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

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

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.

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`

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.

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

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

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

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

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

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.

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

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

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

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

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.

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

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

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

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.

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)`

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

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

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|

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

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

.

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`.

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`

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

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

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

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`

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

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

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

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`

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

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

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

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)`

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.

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

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

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

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`

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.

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.

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.

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

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`.

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`.

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`.

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

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`

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`

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

0

-1

1

3

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

## Solutions for Chapter 10: Vector Algebra

## 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.

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