Scalar and Vector Quantity

What is scalar and vector quantity with 20+ examples? Comparison, Types of vectors, Addition & Subtraction of vectors, Product of vectors and Mock Tests.

In physics, we come across various physical quantities such as distance, mass, velocity, speed, acceleration, momentum, electric current, electric flux, electric field, dipole moment, time, temperature, speed of light and so many.

These physical quantity divided into two main classes:

1. Scalar quantities or Scalars

2. Vector quantities or Vectors

In this article, we discuss all about of scalar and vector quantities. This is useful for all competitive exams, Entrance, TGT PGT (Physics), vectors class 11, vectors class 12. Please read carefully complete article.

Scalar quantities | Scalars

What Is Scalar Quantity?

Those physical quantities which are pass only magnitude

or

Those physical quantities which are completely Defined by only magnitude no required direction in the space, called Scalar quantities.

Examples of Scalar quantities:-

Mass, volume, distance, temperature, pressure, time, speed, electric current, electric flux, magnetic flux, electric potential, work, Energy, power, heat, specific heat, frequency, speed of light, etc.

Vector quantity | Vectors

what are vectors in physics?

Those physical quantities which possess both magnitude and direction and also obey vector algebraic rules.

or

Those physical quantities which can be completely defined by magnitude and direction are called vector quantities.

Examples of vector quantities:-

Position, displacement, velocity, acceleration, momentum, weight, force, torque, impulse, thrust, electric field, magnetic field, gravitational field, electric current density, area, amplitude, wavelength, surface area etc.

Special Note: A quantity having magnitude and direction is not necessarily a vector. For example, time and electric current. These quantities have magnitude and direction but they are scalar. This is because they do not obey the laws of vector addition.

Vector representation | Expression Of vector |How to write vectors?

We know that vectors have both magnitude and direction, so for the Expression of vector use an arrow overhead on the quantity.

\vec{A}=\mid\vec{A}\mid\hat{A}\Rightarrow or \Rightarrow\vec{A}=A\hat{A}

Where,

\vec{A}=vector A

|\vec{A}|=A =magnitude..of..vector A(or\vec{A})

and  = direction of vector A

Tensor Quantity

A physical quantity which has different values in different directions is called a Tensor.

Examples of Tensor: Moment of inertia, refractive index, stress, strain, density etc.

50 Examples of scalar and vector quantities

Distinguish between Scalar and Vector quantity

Difference between scalar and vector quantity| scalar vs vector| 20 examples of scalar and vector quantities

Types of vectors

Depending on the nature of magnitude and direction, several types of vectors

1. Zero vectors | Null vector| Improper vector

Vectors having, Zero magnitudes and Arbitrary (unknown) directions, are called zero vectors.

Notation of zero vector 

\overrightarrow{0}

Special Notes on Null vectors

\overrightarrow{A}+\overrightarrow{0}=\overrightarrow{A}

\overrightarrow{A}×\overrightarrow{0}=\overrightarrow{0}

λ\overrightarrow{0}=\overrightarrow{0}

\overrightarrow{A}=\overrightarrow{B}\Rightarrow\overrightarrow{A}-\overrightarrow{B}=\overrightarrow{0}

If vector A and vector B are parallel to each other then

\overrightarrow{A}×\overrightarrow{B}=\overrightarrow{0}

Examples of null vectors:

• The position vector of a particle at the origin
• The displacement vector of a stationary object
• The acceleration vector of a particle moving with uniform velocity

2. Unit vectors | What are unit vectors?

Vectors having unit magnitude and definite (known) direction, are called unit vectors.

We know that

\overrightarrow{A}=|\overrightarrow{A}|.Â

so that unit vector

Â=\frac{\overrightarrow{A}}{|\overrightarrow{A}|}

Special Notes:

1. Unit vectors gives only direction of vectors i.e. it indicates only direction.
2. Unit vectors have no any unit.

3. Equal vectors

Two vectors are said to be equal when they have equal magnitudes and the same directions and represent the same physical quantity.

vector A and vector B be equal i.e.

\overrightarrow{A}=\overrightarrow{B}

when magnitude

|\overrightarrow{A}|=|\overrightarrow{B}|

and direction

\widehat{A}=\widehat{B}

4. Parallel vectors

Two Vectors A and B be parallel, i.e.

\overrightarrow{A}\parallel\overrightarrow{B}

when

1. Both have the same direction

\widehat{A}=\widehat{B}

2. One vector is scalar (+ve) non-zero multiple of another vector

\overrightarrow{A}=k\overrightarrow{B}

where k is any scalar or number.

5. Antiparallel vectors

Two vectors are said to be anti-parallel, when

1. Both have opposite direction

\widehat{A}=-\widehat{B}

2. One vector is a scalar non-zero (-ve) multiple of another vector.

\overrightarrow{A}=-k\overrightarrow{B}

where k is scalar or any number.

6. Collinear vectors

Those vectors, which act along the same line, are called collinear vectors. So the angle between them can be 0° or 180°.

Special Notes:

1. Every colinear vectors becomes parallel vectors when angle between them be zero but consverse is not true.
2. Every colinear vectors becomes antiparallel vectors when angle between them be 180° but converse is not true.

7. Polar vectors | Radial vectors

Vectors which directly point towards the direction of the vector quantity are called polar vectors.

Examples: Displacement, force, velocity, linear momentum, etc.

8. Axial vectors | Pseudo vectors

These represent rotational effects and are always along the axis of rotation in accordance with the right-handed screw rule.

Examples: Angular velocity, angular momentum, angular acceleration, Torque, etc.

9. Coplanar vectors

what are coplanar vectors? If three or more vectors lie in the same plane, then these are called coplanar vectors.

10. Negative Vectors

A vector of the same magnitude with the opposite direction of the given vector is called the negative vector of that vector.

If\mid\vec{A}\mid=\mid\vec{B}\mid with .. \hat{A}=-\hat{B}

then vectors A & B are said to be negative vectors of each vector.

11. Orthogonal vectors:

When three unit vectors (î, ĵ & k) are formed a right-handed triad then they are called orthogonal unit vectors.

12. Like and Unlike vectors

If the vector representing of a physical quantity has the same direction then they are called, Like vectors.

If they are oppositely directed they are called, Unlike vectors.

13. Concurrent vectors | Coinitial vectors

Vectors have the same origin, called Concurrent vectors or Coinitial vectors.

14. Gradient vectors

A gradient vector is a vector used to represent a vector field. Example: Electric intensity vector.

Addition and Subtraction of two vectors

Addition of two vectors

How to add vectors? If the angle between two non zero vectors A & B is θ then

1. Magnitude of the resultant vectors(R)

1. Direction of the resultant vectors

tan\beta=\frac{CD}{ON}=\frac{Asin\theta}{A+Bcos\theta}

Special Cases :

• R_max = A + B when θ = 0⁰
• R_min = A – B when θ = 180⁰
• R =√ (A² + B²) when θ = 90⁰

Subtraction of two vectors

Direction of resultant vector

tan\alpha2=\frac{Bsin\theta}{A-Bcos\theta}

For complete study♡ of Addition and Subtraction of vectors click here

Multiplication of vectors|Product of two vectors

There are two types of multiplication/product of two vectors_

1. Scalar product or Dot Product
2. Vector Product or Cross Product

1.Dot product of two vectors | The scalar product of two vectors | Direct product of vectors

The dot product of two vectors is defined as the product of the magnitude of two vectors with the cosine of the angle between them.

\overrightarrow{A}.\overrightarrow{B}=ABcos\theta

2. Vector Product | Cross Product of two Vectors | Outer Product

If two vectors A & B having angle θ between them then their cross product is written as

\vec{A}\times\vec{B}\Rightarrow read ...as...\vec{A}cross\vec{B}

\vec{A}\times\vec{B}=[\vec{A},\vec{B}]=ABsin\theta\hat{n}

For Complete study♡ of “Multiplication of vectors” click here

Tips & Tricks about Scalars and vectors

• Similarities between scalar and vector quantity : A quantity having magnitude and direction is not necessarily a vector. For example, time and electric current. These quantities have magnitude and direction but they are scalar. This is because they do not obey the laws of vector addition.

• A vector can have only two rectangular components in-plane and only three rectangular components in space.
• A vector can have any number, even infinite components. (minimum 2 components)
• The rectangular components cannot have magnitude greater than that of the vector itself.

• Distance covered is a scalar quantity.
• The displacement is a vector quantity.
• Scalars are added, subtracted or divided algebraically.
• Vectors are added and subtracted geometrically.
• Division of vectors is not allowed as directions cannot be divided.
• Unit vector gives the direction of vector.
• The magnitude of unit vector is 1.
• Unit vector has no unit.

1. Minimum number of collinear vectors whose resultant can be zero is two.
2. Minimum number of coplanar vectors whose resultant is zero is three.
3. Minimum number of non-coplanar vectors whose resultant is zero is four.

FAQs [Frequantly Ask Questions]

Free Mock test: Introduction of Scalars and Vectors

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