Vector Algebra: Chapter 10 Links
Current page: Vector Algebra 10.1 NCERT Solutions. Go to other exercises and content from this chapter :
10.2 | 10.3 | 10.4 | Misc. | MCQs
Vector Algebra Exercise 10.1 introduces the basic concepts of vectors. In this exercise, you will learn the difference between scalar and vector quantities, directed line segments, position vectors, magnitude of a vector, direction cosines, and the different types of vectors such as zero, unit, coinitial, collinear, equal and negative vectors.
\displaystyle \vec r=\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k},\qquad |\vec r|=\sqrt{x^2+y^2+z^2},\qquad l=\cos\alpha,\;m=\cos\beta,\;n=\cos\gamma
The main idea is to understand what a vector represents, how to write it using standard notation, and how to find its magnitude and direction. You will also become familiar with the different types of vectors and the basic terminology used throughout the chapter.
By practising these questions, you will develop a strong foundation in Vector Algebra, making it easier to understand vector operations such as addition, scalar multiplication, dot product and cross product in the upcoming exercises. These concepts are important for the CBSE board examinations and CUET.
Key Concepts
Before solving Exercise 10.1, it is important to understand the basic concepts of vectors, their notation, magnitude, position vectors, direction cosines and the different types of vectors. These concepts form the foundation for the entire chapter.
1. Scalar and Vector Quantities
A scalar quantity has only magnitude, whereas a vector quantity has both magnitude and direction.
Examples:
- Scalars: Length, mass, time, distance, speed, area, volume, work.
- Vectors: Displacement, velocity, acceleration, force, momentum.
2. Directed Line Segment and Vector
A directed line segment has both magnitude and direction, and is called a vector.
It is denoted by
\displaystyle \overrightarrow{AB}\quad\text{or}\quad\vec a
where A is the initial point and B is the terminal point.
3. Position Vector
If P(x,y,z) is a point with respect to the origin O, then its position vector is
\displaystyle \vec r=\overrightarrow{OP}=x\hat i+y\hat j+z\hat k
The magnitude of the position vector is
\displaystyle |\vec r|=\sqrt{x^2+y^2+z^2}
4. Direction Cosines
If the vector \vec r makes angles \alpha,\beta,\gamma with the positive x, y and z axes respectively, then
\displaystyle l=\cos\alpha,\qquad m=\cos\beta,\qquad n=\cos\gamma
These are called the direction cosines of the vector.
Also,
\displaystyle l^2+m^2+n^2=1
5. Types of Vectors
- Zero Vector: Magnitude is zero.
- Unit Vector: Magnitude is 1 and is denoted by \hat a.
- Coinitial Vectors: Vectors having the same initial point.
- Coterminus Vectors: Vectors having the same terminal point are called coterminus vectors.
- Collinear Vectors: Vectors parallel to the same line.
- Equal Vectors: Same magnitude and same direction.
- Like and Unlike Vectors: Vectors having the same direction are called like vectors, while vectors having opposite directions are called unlike vectors. They may or may not have the same magnitude.
- Negative of a Vector: The negative of \vec a is -\vec a. It has the same magnitude as \vec a but acts in the opposite direction.
6. Important Points
- Magnitude of a vector is always non-negative.
- A zero vector has magnitude zero and no definite direction.
- Position vectors are always measured from the origin.
- Direction cosines satisfy l^2+m^2+n^2=1.
- Equal vectors need not have the same initial point.
- The negative of \vec a is -\vec a. Both have the same magnitude but opposite directions.
- Line of Support: The straight line of which the directed line segment representing a vector forms a part is called the line of support of the vector. A vector can be shifted parallel to itself without changing its magnitude or direction, provided its line of support remains parallel.
Quick Reference Table
| Concept | Details |
|---|---|
| Scalar Quantity |
Has magnitude only. Examples: Length, mass, time, distance, speed. |
| Vector Quantity |
Has magnitude and direction. Examples: Displacement, velocity, acceleration, force. |
| Position Vector | \displaystyle \vec r=\overrightarrow{OP}=x\hat i+y\hat j+z\hat k \displaystyle |\vec r|=\sqrt{x^2+y^2+z^2} |
| Direction Cosines | \displaystyle l=\cos\alpha,\quad m=\cos\beta,\quad n=\cos\gamma \displaystyle l^2+m^2+n^2=1 |
| Unit Vector |
Magnitude is 1. \displaystyle \hat a=\frac{\vec a}{|\vec a|} |
| Types of Vectors |
Zero, Unit, Coinitial, Collinear, Equal and Negative of a vector. |
Before You Begin… Do not rely only on the magnitude of a vector. Always pay attention to its direction, notation and the exact definition being used in the question.
Let us now solve all the NCERT questions of Vector Algebra 10.1 step-by-step.
Question 1: Graphical Representation
1. Represent graphically a displacement of 40 km, 30° east of north.
Solution
Think First… A vector is completely specified only when both its magnitude and direction are given. Here, the magnitude is 40 km and the direction is 30° east of north. So, first draw the north direction, then measure an angle of 30° towards the east, and finally represent the displacement by a directed line segment.
We have,
- Magnitude of displacement = 40 km
- Direction = 30° east of north
Choose a convenient scale.
For example, let
Scale: 1 cm = 10 km
Then, the required displacement is represented by a directed line segment of length 4 cm.
Starting from a point O, draw the north direction. From it, measure an angle of 30° towards the east and draw a 4 cm long directed line segment in that direction, as shown below.
The directed line segment \displaystyle \overrightarrow{OP} represents a displacement of 40 km, 30° east of north.
Hence, the required graphical representation is shown above.
Question 2: Scalar and Vector Quantities
2. 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
Solution
Think First… A quantity having only magnitude is a scalar, whereas a quantity having both magnitude and direction is a vector.
Therefore,
- (i) 10 kg – Scalar (Mass has magnitude only.)
- (ii) 2 metres north-west – Vector (Magnitude and direction are both given.)
- (iii) 40° – Scalar (Angle has magnitude only.)
- (iv) 40 watt – Scalar (Power has magnitude only.)
- (v) 10^{-19} coulomb – Scalar (Electric charge has magnitude only.)
- (vi) 20 m/s² – Vector (Acceleration is a vector quantity.)
You may already be following Maths Better for the NCERT Solutions of the previous Class 12 Maths chapters, namely:
- Matrices
- Determinants
- Relations and Functions
- Inverse Trigonometric Functions
- Continuity and Differentiability
- Application of Derivatives
- Integrals
- Application of Integrals
- Differential Equations
Likewise, Vector Algebra 10.1 builds the foundation of the entire chapter by introducing the basic concepts of vectors. Once you clearly understand scalar and vector quantities, graphical representation of vectors, and the properties of coinitial, equal and collinear vectors, the vector operations covered in the remaining exercises will become much easier. So, take your time to master these fundamentals before moving ahead. Now, let’s proceed to the next question.
Question 3: Vector Algebra 10.1
3. Classify the following as scalar and vector quantities.
(i) Time period (ii) Distance (iii) Force (iv) Velocity (v) Work done
Solution
Think First… Physical quantities having only magnitude are scalars, whereas those having both magnitude and direction are vectors.
Therefore,
- (i) Time period – Scalar (It has magnitude only.)
- (ii) Distance – Scalar (It has magnitude only.)
- (iii) Force – Vector (It has both magnitude and direction.)
- (iv) Velocity – Vector (It has both magnitude and direction.)
- (v) Work done – Scalar (It has magnitude only.)
Question 4: Vector Algebra 10.1
4. In the given figure (a square), identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
Solution
Think First… Coinitial vectors have the same initial point. Equal vectors have the same magnitude and the same direction. Collinear vectors are parallel to the same line, but they need not have the same direction.
From the given figure, we observe the following.
(i) Coinitial vectors
The vectors \vec a and \vec d have the same initial point.
Therefore, \vec a and \vec d are coinitial vectors.
(ii) Equal vectors
The vectors \vec b and \vec d have the same magnitude and the same direction.
Therefore, \vec b and \vec d are equal vectors.
(iii) Collinear but not equal vectors
The vectors \vec a and \vec c are parallel to the same line but act in opposite directions.
Therefore, \vec a and \vec c are collinear but not equal vectors.
By now, you have already completed the NCERT Solutions for Matrices, Determinants, Relations and Functions, Inverse Trigonometric Functions, Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals and Differential Equations.
Likewise, every exercise of Vector Algebra is explained here with detailed, step-by-step solutions to help you develop a clear understanding of vector concepts and prepare confidently for your CBSE board examinations.
Many of these questions are also available in video format on my YouTube Channel, @MathsBetter, where each solution is explained in a simple and student-friendly manner. Now, let’s proceed to the next question.
Question 5: Vector Algebra 10.1
5. Answer the following as True or False.
Solution
Think First… Equal vectors must have the same magnitude and the same direction. Collinear vectors are parallel to the same line, but they may have different magnitudes or opposite directions.
(i) \vec a and -\vec a are collinear.
Answer: True
Reason: \vec a and -\vec a have the same line of support but opposite directions. Therefore, they are collinear.
(ii) Two collinear vectors are always equal in magnitude.
Answer: False
Reason: Collinear vectors may have different magnitudes. Being collinear only means that they are parallel to the same line.
(iii) Two vectors having the same magnitude are collinear.
Answer: False
Reason: Two vectors may have the same magnitude but different directions. Hence, they need not be collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Answer: False
Reason: Equal vectors must have the same magnitude as well as the same direction. Two collinear vectors may have opposite directions and, therefore, need not be equal.
Maths Better Tip! Understanding Collinear Vectors: Two vectors need not have the same line of support to be collinear. If their lines of support are the same or are parallel to the same straight line, the vectors are called collinear vectors. Later in this chapter, after learning about the components of vectors and scalar multiplication, we shall see that two non-zero vectors are collinear if one is a scalar multiple of the other. For now, focus on the geometric meaning of collinear vectors.
Common Mistakes to Avoid
- Confusing scalars and vectors: Remember that a scalar has only magnitude, whereas a vector has both magnitude and direction.
- Ignoring the direction while drawing a vector: A vector is represented by a directed line segment. The arrowhead is essential and indicates the direction of the vector.
- Using an unsuitable scale: While representing vectors graphically, choose a convenient scale and use it consistently throughout the figure.
- Confusing equal vectors with collinear vectors: Equal vectors must have the same magnitude and the same direction, whereas collinear vectors only need to be parallel to the same line.
- Confusing coinitial vectors with equal vectors: Coinitial vectors have the same initial point, but they need not have the same magnitude or direction.
- Misunderstanding the negative of a vector: The negative of a vector has the same magnitude as the given vector but acts in the opposite direction.
- Assuming vectors with the same magnitude are equal: Two vectors are equal only if they have both the same magnitude and the same direction.
- Confusing distance with displacement: Distance is a scalar quantity, whereas displacement is a vector quantity because it also specifies direction.
- Assuming all physical quantities are vectors: Quantities such as mass, time, work and power are scalars, while displacement, velocity, force and acceleration are vectors.
- Ignoring the direction while answering conceptual questions: In vector algebra, direction is as important as magnitude. Always check both before concluding whether two vectors are equal, opposite or collinear.
Continue Learning
In Vector Algebra 10.1, we learned the basic concepts of Vectors. This exercise includes identifying scalar and vector quantities, representing vectors graphically, understanding coinitial, equal and collinear vectors, and answering conceptual questions based on the properties of vectors.
Before starting Exercise 10.2, revise the following concepts:
- The difference between scalar and vector quantities, and examples of each.
- How a vector is represented graphically using a directed line segment, including the importance of both magnitude and direction.
- How to choose a suitable scale while drawing vectors to represent the given magnitude.
- The meaning of coinitial vectors, equal vectors, collinear vectors and negative of a vector.
- The conditions under which two vectors are said to be equal, namely the same magnitude and the same direction.
- The difference between distance and displacement, as well as between speed and velocity.
- Common physical quantities such as mass, time, work and power (scalars), and displacement, velocity, force and acceleration (vectors).
- The importance of checking both magnitude and direction before deciding whether two vectors are equal, opposite or merely collinear.
Explore More
A clear understanding of the basic concepts of Vector Algebra forms the foundation for the entire chapter. Learn to distinguish between scalar and vector quantities, identify coinitial, equal and collinear vectors correctly, and remember that a vector is completely determined by both its magnitude and direction. While representing vectors graphically, always choose a suitable scale and draw the direction carefully. In conceptual questions, avoid making conclusions based only on magnitude; always check the direction as well. With these fundamental concepts clear, you will find the remaining exercises on vectors much easier to understand and solve.
All the best and keep learning 👍
