NCERT Solution for Class 12 Mathematics Chapter 10, “Vector Algebra,” is an important study material designed to help students understand vector algebra’s fundamental concepts and principles. Swastik Classes, a leading coaching institute, has developed comprehensive NCERT solutions that provide step-by-step explanations and solved examples to help students develop a deeper understanding of the subject. The chapter covers the scalar and vector products of two and three-dimensional vectors, triple scalar product, and vector triple product. With the help of Swastik Classes’ NCERT solutions, students can improve their problem-solving skills and gain the confidence to tackle complex vector algebra problems. These solutions are also helpful for students who are preparing for competitive exams like JEE, NEET, and other entrance exams. Overall, Swastik Classes’ NCERT Solution for Class 12 Mathematics Chapter 10 is an essential resource for students who want to excel in mathematics and build a strong foundation in vector algebra.

## Download PDF of NCERT Solutions for Class 12 Mathematics Chapter 10 vector

**Answers of Mathematics NCERT solutions for class 12 Chapter 10 Vector**

**Exercise 10.1**

**Question 1:**

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

**Answer:**

Here, vector represents the displacement of 40 km, 30° East of North.

**Question 2:**

Classify the following measures as scalars and vectors.

- 10 kg
- 2 metres north-west
- 40°
- 40 Watt
- 10
^{–19}coulomb - 20 m/s
^{2}

**Answer: **

(i) 10 kg is a scalar quantity because it involves only magnitude.

(ii) 2 meters north-west is a vector quantity as it involves both magnitude and direction.

(iii) 40° is a scalar quantity as it involves only magnitude.

(iv) 40 watts is a scalar quantity as it involves only magnitude.

(v) 10^{–19} coulomb is a scalar quantity as it involves only magnitude.

(vi) 20 m/s^{2} is a vector quantity as it involves magnitude as well as direction.

**Question 3:**

Classify the following as scalar and vector quantities.

- time period
- distance
- force
- velocity
- work done

**Answer: **

(i) Time period is a scalar quantity as it involves only magnitude.

(ii) Distance is a scalar quantity as it involves only magnitude.

(iii) Force is a vector quantity as it involves both magnitude and direction.

(iv) Velocity is a vector quantity as it involves both magnitude as well as direction.

(v) Work done is a scalar quantity as it involves only magnitude.

**Question 4:**

In Figure, identify the following vectors.

(i) Co initial (ii) Equal (iii) Collinear but not equal

**Answer: **

(i) Vectors _{}and _{} are co initial because they have the same initial point

(ii) Vectors _{} and _{} are equal because they have the same magnitude and direction.

(iii) Vectors _{} and _{} are collinear but not equal. This is because although they areparallel, their directions are not the same.

**Question 5:**

Answer the following as true or false.

(i) _{}and _{} are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

**Answer: **

(i) True.

Vectors _{}and _{} are parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

Two vectors having the same magnitude need not necessarily be parallel to the same line.

(iv) False

Only if the magnitude and direction of two vectors are the same, regardless of the positions of their initial points the two vector are said to be equal.

**Exercise 10.2**

**Question 1:**

Compute the magnitude of the following vectors:

**Answer:**

The given vectors are:

**Question 2:**

Write two different vectors having same magnitude.

**Answer:**

Hence, are two different vectors having the same magnitude. The vectors are different because they have different directions.

**Question 3:**

Write two different vectors having same direction.

**Answer: **

The direction cosines of are the same. Hence, the two vectors have the same direction.

**Question 4:**

Find the values of x and y so that the vectors are equal

**Answer:**

The two vectors will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

**Question 5:**

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

**Answer:**

The vector with the initial point P(2,1) and terminal point Q(–5,7) can be given by,

Hence, the required scalar components are–7 and 6 while the vector components are _{} and _{}

**Question 6:**

Find the sum of the vectors .

**Answer**

The given vectors are .

**Question 7:**

Find the unit vector in the direction of the vector .

**Answer**

The unit vector _{} in the direction of vector _{} is given by _{}

**Question 8:**

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

**Answer**

The given points are P (1, 2, 3) and Q (4, 5, 6).

Hence, the unit vector in the direction ofis

**Question 9:**

For given vectors, and , find the unit vector in the direction of the vector

**Answer**

The given vectors are and .

Hence, the unit vector in the direction of _{} is

.

**Question 10:**

Find a vector in the direction of vector which has magnitude 8units.

**Answer**

Hence, the vector in the direction of vector which has magnitude 8 units is given by,

**Question 11:**

Show that the vectors are collinear.

**Answer**

.

Hence, the given vectors are collinear.

**Question 12:**

Find the direction cosines of the vector _{}

**Answer**

Hence, the direction cosines of _{} are _{}

**Question 13:**

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

**Answer:**

The given points are A (1, 2, –3) and B (–1, –2, 1).

Hence, the direction cosines of are

**Question 14:**

Show that the vector _{} is equally inclined to the axes OX, OY, and OZ.

**Answer:**

Therefore, the direction cosines of _{}

Now, letα, β, and γ be the angles formed by with the positive directions of x, y, and z axes.

Then, we have _{}

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

***Question 15:**

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, in the ration2:1internallyexternally

**Answer:**

The position vector of point R dividing the line segment joining two points P and Q in the ratio m: n is given by:

Internally:

Externally:

Position vectors of P and Q are given as:

ThepositionvectorofpointRwhichdividesthelinejoiningtwopointsPandQ internally in the ratio 2:1 is given by,

The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,

***Question 16:**

Find the position vector of the mid-point of the vector joining the points

P (2, 3, 4) and Q (4, 1, – 2).

**Answer**

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, –2) is given by,

***Question 17:**

Show that the points A, B and C with position vectors, _{},

_{} respectively form the vertices of a right-angled triangle.

**Answer:**

Position vectors of points A, B, and C are respectively given as:

Hence, ABC is a right-angled triangle.

***Question 18:**

In triangle ABC which of the following is not true:

A.

B.

C.

D.

**Answer:**

On applying the triangle law of addition in the given triangle, we have:

From equations (1) and (3), we have:

Hence, the equation given in alternative C is incorrect. The correct answer is C.

***Question 19:**

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

A. , for some scalar λ

B.

C. the respective components of are proportional

D. both the vectors have same direction, but different magnitudes

**Answer:**

If are two collinear vectors, then they are parallel. Therefore, we have:

(For some scalar λ)

If λ = ±1, then.

Thus, the respective components of are proportional. However, vectors can have different directions.

Hence, the statement given in D is incorrect.

The correct answer is D.

**Exercise 10.3**

**Question 1:**

Find the angle between two vectors and with magnitudes and 2, respectively having .

**Answer:**

It is given that,

Hence, the angel between the given vectors _{} and _{} is _{}

**Question 2:**

Find the angle between the vectors _{}

**Answer:**

The given vectors are _{} and _{}

Also, we know that _{}

**Question 3:**

Find the projection of the vector _{} on the vector _{}.

**Answer: **

Let and _{}.

Now, projection of vector on is given by,

Hence, the projection of vector on is 0.

**Question 4:**

Find the projection of the vector on the vector .

**Answer:**

Let and .

Now, projection of vector on is given by,

**Question 5:**

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

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

**Answer:**

Thus, each of the given three vectors is a unit vector.

Hence, the given three vectors are mutually perpendicular to each other.

**Question 6:**

Find _{} and _{}, if _{}.

**Answer**

**Question 7:**

Evaluate the product _{}.

**Answer**

**Question 8:**

Find the magnitude of two vectors , having the same magnitude and such that the angle between them is 60° and their scalar product is _{}.

**Answer**

Let θ be the angle between the vectors It is given that

We know that

**Question 9:**

Find _{}, if for a unit vector _{}.

**Answer**

**Question 10:**

If are such that is perpendicular to _{}, then find the value of λ.

**Answer:**

Hence, the required value of λ is 8.

**Question 11:**

Show that _{} is perpendicular to _{}, for any two nonzero vectors _{}

**Answer:**

Hence, _{}and _{} are perpendicular to each other.

**Question 12:**

If, then what can be concluded about the vector ?

**Answer: **

It is given that.

Hence, vector satisfying can be any vector.

***Question 13:**

If _{} and _{} are unit vectors such that _{} find the value of _{}

**Answer: **

Consider the given vectors,

Given _{}

So,

_{} [Distributivity of scalar product over addition]

_{} …..(1) _{}

Next

_{} ….(2) _{}

And,

_{} …..(3) _{}

From (1), (2) and (3),

_{} [Scalar product is commutative]

Hence the value is _{}

***Question 14:**

If either vector , then . But the converse need not be true. Justify your answer with an example.

**Answer:**

We now observe that:

Hence, the converse of the given statement need not be true.

***Question 15:**

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 and _{}]

**Answer:**

The vertices of OABC are given as A(1,2,3), B(–1,0,0), and C (0,1,2). Also, it is given that ∠ABC is the angle between the vectors and .

Now, it is known that:

.

***Question 16:**

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

**Answer:**

The given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

Hence, the given points A, B, and C are collinear.

***Question 17:**

Show that the vectors form the vertices of a right-angled triangle.

**Answer:**

Let vectors be position vectors of points A, B, and C respectively.

Now, vectors represent the sides of OABC.

Hence, OABC is a right-angled triangle.

***Question 18:**

If _{} is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λis unit vector if

(A) λ = 1

(B) λ = –1

(C) _{}

(D) _{}

**Answer:**

Vector _{} is a unit vector if _{}

Hence, vector _{} is a unit vector if _{} The correct answer is D.

**Exercise 10.4**

**Question 1:**

Find _{} if _{} and _{}.

**Answer:**

We have, _{}

and _{}

**Question 2:**

Find a unit vector perpendicular to each of the vector and , where_{} and _{}.

**Answer:**

We have,

_{} and _{}

Hence, the unit vector perpendicular to each of the vectors and is given by the relation,

**Question 3:**

If a unit vector _{} makes an angles _{} with _{} with _{} and an acute angle θ with _{} then find θ and hence, the compounds of _{}

**Answer:**

Let unit vector have (a_{1}, a_{2}, a_{3}) components.

Since is a unit vector, _{}.

Also, it is given that _{} makes an angles _{} with _{} with _{} and an acute angle θ with _{} then we have:

Hence, _{} and the components _{} of are _{}

**Question 4:**

Show that

**Answer:**

**Question 5:**

Find λ and µ if

**Answer:**

On comparing the corresponding components, we have:

Hence, _{}

**Question 6:**

Given that and . What can you conclude about the vectors ?

**Answer**

Then,

Either _{} or _{}, or _{}

Either _{} or _{}, or _{}. But, and cannot be perpendicular and parallel simultaneously.

Hence, or .

***Question 7:**

Let the vectors given as _{}. Then show that _{}

**Answer:**

We have,

On adding (2) and (3), we get:

Now, from (1) and (4), we have:

Hence, the given result is proved.

***Question 8:**

If either or , then Is the converse true? Justify your answer with an example.

**Answer:**

Take any parallel non-zero vectors so that.

It can now be observed that:

Hence, the converse of the given statement need not be true.

***Question 9:**

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

C (1, 5, 5).

**Answer**

The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and

C (1, 5, 5).

The adjacent sides and of OABC are given as:

Hence, the area of OABC

***Question 10:**

Find the area of the parallelogram whose adjacent sides are determined by the vector

**Answer**

The area of the parallelogram whose adjacent sides are _{} is _{} Adjacent sides are given as:

Hence, the area of the given parallelogram is .

***Question 11:**

Let the vectors _{}and _{} be such that _{} and _{} then, _{} is a unit vectors, if the angle between _{}and _{} is

(A) _{} (B) _{} (C) _{} (D) _{}_{ }_{}

**Answer**

It is given that _{} and _{}

We know that _{}, where _{} is a unit vector perpendicular to both. _{} and _{} and θ is the angle between _{} and _{}

Now _{} is a unit vector if _{}

Hence, _{} is a unit vector if the angle between _{}and _{} is _{}. The correct answer is B.

***Question 12:**

Area of a rectangle having vertices A, B, C, and D with position vectors

_{} and _{} respectively is

(A) _{} (B) 1 (C) 2 (D) 4

**Answer:**

The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:

The adjacent sides and of the given rectangle are given as:

Now, it is known that the area of a parallelogram whose adjacent sides are is _{}.

Hence, the area of the given rectangle is _{} The correct answer is C.

**Miscellaneous Solutions**

**Question 1:**

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

**Answer:**

If is a unit vector in the XY-plane, then

Here, θ is the angle made by the unit vector with the positive direction of the x-axis. Therefore, for θ = 30°:

Hence, the required unit vector is _{}.

**Question 2:**

Find the scalar components and magnitude of the vector joining the points

.

**Answer:**

The vector joining the points _{} can be obtained by,

Hence, the scalar components and the magnitude of the vector joining the given points are respectively

_{} and .

**Question 3:**

A 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.

**Answer:**

Let O and B be the initial and final positions of the girl respectively. Then, the girl’s position can be shown as:

Now, we have:

By the triangle law of vector addition, we have:

Hence, the girl’s displacement from her initial point of departure is

**Question 4:**

**If **_{} then is it true that _{} Justify your answer.

**Answer**

Now, by the triangle law of vector addition, we have .

It is clearly known that _{} represent the sides of OABC.

Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side.

Hence, it is not true that _{}.

**Question 5:**

Find the value of x for which _{} is a unit vector.

**Answer**

_{} is a unit vector _{}

Hence, the required value of x is _{}

**Question 6:**

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

**Answer:**

We have,

Let _{} be the resultant of _{}

Hence, the vector of magnitude 5 units and parallel to the resultant of vectors is

**Question 7:**

If , find a unit vector parallel to the

Vector .

**Answer:**

We have,

Hence, the unit vector along is

**Question 8:**

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.

**Answer:**

The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).

Thus, the given points A, B, and C are collinear.

Now, let point B divide AC in the ratio . Then, we have:

On equating the corresponding components, we get:

Hence, point B divides AC in the ratio

**Question 9:**

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

**Answer:**

It is given that

.

It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:

Therefore, the position vector of point R is . Position vector of the mid-point of RQ=

Hence, P is the mid-point of the line segment RQ.

**Question 10:**

The two adjacent sides of a parallelogram are and . Find the unit vector parallel to its diagonal. Also, find its area.

**Answer**

Adjacent sides of a parallelogram are given as: and _{}Then, the diagonal of a parallelogram is given by .

Thus, the unit vector parallel to the diagonal is

Hence, the area of the parallelogram is square units.

***Question 11:**

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are _{}

**Answer**

Let a vector be equally inclined to axes OX, OY, and OZ at angle α. Then, the direction cosines of the vector are cos α, cos α, and cos α.

Hence, the direction cosines of the vector which are equally inclined to the axes are

***Question 12:**

Let and . Find a vector _{} which is perpendicular to both and , and.

**Answer**

Let

.

Since _{}is perpendicular to both _{}, we have:

Also, it is given that:

On solving (i), (ii), and (iii), we get:

Hence, the required vector is_{}.

***Question 13:**

The scalar product of the vector with a unit vector along the sum of vectors

_{} and _{} is equal to one. Find the value of _{}

**Answer**

Scalar product of _{} with this unit vector is 1.

Hence, the value of λ is 1.

***Question 14:**

If _{} are mutually perpendicular vectors of equal magnitudes, show that the vector _{} is equally inclined to _{}

**Answer**

Since are mutually perpendicular vectors, we have

It is given that:

Let vector be inclined to at angles _{} respectively. Then, we have:

Now, as _{}, .

Hence, the vector _{} is equally inclined _{}

***Question 15:**

Prove that _{}, if and only if _{} are perpendicular, given _{}

**Answer:**

***Question 16:**

If θ is the angle between two vectors and , then only when

(A) _{} (B) _{} (C) (D)

**Answer:**

Let θ be the angle between two vectors and.

Then, without loss of generality, and are non-zero vectors so that

Hence _{} when _{} The correct answer is B.

***Question 17:**

Let and be two unit vectors and θ is the angle between them. Then is a unit vector if

(A) _{} (B) _{}

(C) _{} (D) _{}

**Answer:**

Let and be two unit vectors and θ be the angle between them.

Then,

Now, is a unit vector if

Hence, is a unit vector if _{}. The correct answer is D.

***Question 18:**

The value of _{} is

(A) 0

(B) –1

(C) 1

(D) 3

**Answer:**

The correct answer is C.

***Question 19:**

If θ is the angle between any two vectors _{} and _{}, then _{} when θ is equal to

(A) 0 (B) _{} (C) _{} (D) _{}

**Answer:**

## Conclusion

Swastik Classes’ NCERT Solution for Class 12 Mathematics Chapter 10, “Vector Algebra,” is a comprehensive study material designed to help students understand the fundamental concepts and principles of vector algebra. The solutions provide step-by-step explanations and solved examples that help students to develop a deeper understanding of the subject. The chapter covers a range of topics including scalar and vector products of two and three-dimensional vectors, triple scalar product, and vector triple product. With the help of these solutions, students can improve their problem-solving skills and gain the confidence to tackle complex vector algebra problems. Swastik Classes’ NCERT solutions are designed in accordance with the latest CBSE syllabus, making them useful for students preparing for board exams or competitive exams like JEE and NEET. Overall, Swastik Classes’ NCERT Solution for Class 12 Mathematics Chapter 10 is an excellent resource for students who want to excel in mathematics and build a strong foundation in vector algebra.

The finest thing the NCERT answers Chapter 10 vector algebra is that it conveys tough parts in vernacular and easy language so that students of all intellect levels can understand them.

### Topics included on NCERT Class 12 Maths Chapter 10

Section no. | Topics |

10.1 | Introduction to Vectors |

10.2 | Some Basic Concepts |

10.3 | Types of Vectors |

10.4 | Addition of Vectors |

10.5 | Multiplication of a Vector by a Scalar |

10.6 | Product of Two Vectors |

### Weightage of Math Class 12 Chapter 10 in CBSE Exam

Chapters | Marks |

Vector Algebra | 5 Marks |

### Related Links

### Videos on Vector Algebra

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## FAQS on NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

### What is meant by vector algebra?

an algebra in which the components involved might be vectors and the assumptions and rules are based on vector behaviour.

### How many exercises are in vector algebra?

5 exercises

### What are 3 types of vectors?

- Null Vector or Zero Vector

When the magnitude of a vector is 0 and the starting point and terminus of the vector are the same, the vector is said to be a Zero Vector. PQ, for example, is a line segment in which the coordinates of point P and point Q are the same. The symbol for a zero vector is 0. There is no fixed orientation for the zero vector.

- Vector of a unit

When the magnitude of a vector is 1 unit in length, it is said to be a unit vector. If x is a vector of magnitude x, then the unit vector is denoted by x in the vector’s direction and has a magnitude equal to 1.

However, two-unit vectors cannot be equal since their directions may differ.

- Travelling position

A position vector is defined as a point X in the plane. It just indicates the current location. Assume OX is a point in a plane with respect to its origin. If O is used as the reference origin and X is an arbitrary point in the plane, the vector is referred to as the point’s position vector.

### What is an equal vector?

When the magnitude and direction of two or more vectors are the same, they are said to be equal.

### What is the triangular law of addition?

The triangle rule of vector addition asserts that when two vectors are represented as two sides of a triangle with the same order of magnitude and direction, the magnitude and direction of the resulting vector is represented by the third side of the triangle.

### What is the resultant of vector?

The vector sum of two or more vectors is the outcome. It’s the outcome of combining two or more vectors. When you put the displacement vectors A, B, and C together, you obtain vector R. Vector R may be found using an appropriately drawn, scaled vector addition diagram, as illustrated in the picture.