Scalar and Vector Quantities: Class 11 Notes

A scalar quantity in physics is a description of magnitude.

It consists of a single number and its corresponding unit of measurement. 

You can consider scalars as simple numerical values that answer the question "how much?".  This does not require you to tell which direction you are headed. 

Key Characteristics of Scalars

1. Definition: Possesses magnitude only.
2. Specification: Always defined by a number and a unit (e.g., 10 kg, 30°C)

Examples of Scalar Quantities

• Distance: The total path length covered between two points (e.g., 5 km).
• Mass: How much matter an object has (e.g., 70 kg).
• Temperature: How hot or cold a body is (e.g., 25 degrees Celsius).
• Time: How long is one event (e.g., 60 seconds).

Scalar Algebra

Scalar quantities follow basic arithmetic when it comes to combining them. 

• Addition and Subtraction: You can add or subtract scalars. The only condition for that is that they must have the same units. For instance, you can add 5 kg and 2 kg to get a total of 7 kg. But it’s impossible to add 5 kg to 10 meters.
• Multiplication and Division: Multiplication and division are also possible with scalars. They don't have to have the same units. For instance, you can calculate density by dividing mass by volume.

 

A vector quantity has magnitude and a specific direction in space. 

This quantity in physics tells us that, in addition to knowing 'how much', we should also know 'which way'. 

Key Characteristics of Vectors

• Definition: It must have magnitude and direction.
• Vector Addition: Vectors are ruled by specific laws. Typically, they follow the triangle law of addition or the parallelogram law of addition. They are different from simple arithmetic.

Triangle Law of Addition - This applies when we have two vectors that are placed head-to-tail. Their sum is the vector from the tail of the first to the head of the second.

Parallelogram Law of Addition - When two vectors form adjacent sides of a parallelogram, their sum is the diagonal vector from their common origin.

Examples of Vector Quantities

• Displacement: The change in position in a specific direction.
• Velocity: The rate of change of displacement (speed in a direction).
• Acceleration: The rate of change of velocity.
• Force: A push or pull on an object in a specific direction.

How to Represent Vectors

To differentiate vectors from scalars in physics, we use specific notations. 

• In typed text: Vectors are represented using boldface letters such as v, a, and F.
• In handwritten work: Vectors are denoted by placing an arrow above the letter, written as , $\stackrel{\to }{v}$ , $\stackrel{\to }{a}$ and $\stackrel{\to }{F}$ .
• Magnitude Notation - The magnitude (or length) of a vector is always a positive scalar value. There are two standard ways to express vector magnitude.

• Using the same letter in regular (non-bold) typeface: the magnitude of v is written as v
• Using absolute value notation: the magnitude of v can also be written as |v|

Both notations represent the same concept. It's the scalar distance or size of the vector quantity.

 

Vectors essentially describe motion and position in space. Here are two fundamental applications.

Position and Displacement Vectors

To describe an object's location, we first need a frame of reference. Typically, we can go for a coordinate system with an origin point (O).

Position Vector (r): This is a vector that extends from the origin (O) to the object's current location (P). It tells us exactly where the object is relative to our reference point.

Displacement Vector (Δr): If an object moves from an initial position (P) to a final position (P'), the displacement vector is the straight-line vector pointing from P to P'.

A displacement vector only depends on the initial and final positions. It’s independent of the actual path taken. Because of this, the magnitude of the displacement is always less than or equal to the total path length (distance) travelled.

Equality of Vectors

Often, you will be asked in annuals and competitive tests, such as the JEE Main, to solve problems regarding the equality of vectors. It’s better to understand the conditions behind it. 

Two vectors, A and B, are considered equal if and only if they meet two conditions. 

• They have the same magnitude (|A| = |B|).
• They have the same direction.

Knowing this, a vector's starting location does not define it. 

You can move a vector parallel to itself anywhere in space, and it remains the same vector. Such vectors are known as free vectors.

Additionally, explore the curated NCERT Solutions for Motion in a Plane. 

 

For quick reference, here are the differences between scalar and vector quantities. 

Basis for Comparison	Scalar Quantities	Vector Quantities
Definition	A quantity with magnitude only.	A quantity with both magnitude and direction.
Specification	A single number and a unit.	A number, a unit, and a direction.
Algebra	Follows ordinary algebra.	Follows vector algebra (e.g., Triangle Law).
Examples	Mass, Speed, Distance, Temperature, Time.	Displacement, Velocity, Acceleration, Force.

Here are some JEE style questions you can prepare when it comes to scalars and vectors. Before checking them, just follow along the few points below. 

• Addition and subtraction are commutative and associative.
• Dot product is zero for perpendicular vectors; cross product is zero for parallel vectors.
• Unit vectors satisfy $\hat{i}\cdot \hat{i}=1,\hat{i}\cdot \hat{j}=0,\hat{i}\times \hat{j}=\hat{k}$ .

Example 1: Two vectors $\vec{A}=3\hat{i}+4\hat{j}$  and $\vec{B}=5\hat{i}-2\hat{j}$ represent displacements. Find the magnitude of the resultant displacement. 

Resultant: $\vec{R}=\vec{A}+\vec{B}=(3+5)\hat{i}+(4-2)\hat{j}=$ $8\hat{i}+2\hat{j}$ . Magnitude: $\left|\vec{R}\right|=\sqrt[]{8^2+2^2}=\sqrt[]{64+4}=\sqrt[]{68}\approx 8.25\mathrm{m}$ .

Example 2: Find the angle between vectors $\vec{A}=2\hat{i}+3\hat{j}$  and $\vec{B}=-\hat{i}+\hat{j}$ using the dot product.

Dot product: $\vec{A}\cdot \vec{B}=(2\cdot -1)+(3\cdot 1)=-2+3=1$ . Magnitudes: $\left|\vec{A}\right|=\sqrt[]{2^2+3^2}=$   $\sqrt[]{13},\left|\vec{B}\right|=\sqrt[]{(-1{)}^2+1^2}=\sqrt[]{2}$ . Angle: $\mathrm{c}\mathrm{o}\mathrm{s}\theta =\frac{\vec{A}\cdot \vec{B}}{\left|\vec{A}\right|\left|\vec{B}\right|}=\frac{1}{\sqrt[]{13}\cdot \sqrt[]{2}}⟹\theta ={\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(\frac{1}{\sqrt[]{26}}\right)\approx {78.7}^{\circ }$ .

Example 3: A boat moves at $5\mathrm{m}/\mathrm{s}$  across a river flowing at $3\mathrm{m}/\mathrm{s}$ . Find the magnitude of the boat's velocity relative to the ground.

Boat's velocity: ${\vec{v}}_{AB}=5\hat{j}$ , river's velocity: ${\vec{v}}_B=3\hat{i}$ . Resultant: ${\vec{v}}_A={\vec{v}}_{AB}+{\vec{v}}_B=3\hat{i}+5\hat{j}$ . Magnitude: $\left|{\vec{v}}_A\right|=\sqrt[]{3^2+5^2}=\sqrt[]{34}\approx 5.83\mathrm{m}/\mathrm{s}$ .

Click on the chapters for Physics and find detailed guides. 

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

Some more Science stream notes for Class 11 CBSE to check.

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes

 

Find the NCERT Solutions for all the chapters in Class 11 right here.   

Units and Measurements Class 11 NCERT Solutions	Mechanical Properties of Solids Class 11 NCERT Solutions
Motion in a Straight Line Class 11 NCERT Solutions	Mechanical Properties of Fluids Class 11 NCERT Solutions
Motion in a Plane Class 11 NCERT Solutions	Thermal Properties of Matter Class 11 NCERT Solutions
Laws of Motion Class 11 NCERT Solutions	Thermodynamics Class 11 NCERT Solutions
Work, Energy, and Power Class 11 NCERT Solutions	Kinetic Theory Class 11 NCERT Solutions
System of Particles and Rotational Motion Class 11 NCERT Solutions	Oscillations Class 11 NCERT Solutions
Gravitation Class 11 NCERT Solutions	Waves Class 11 NCERT Solutions

Keep exploring the regularly updated NCERT Solutions for Physics Class 11!