SCALARS&

VECTORS

What is Scalar?

Length of a car is 4.5 mphysical quantity magnitude

Mass of gold bar is 1 kgphysical quantity magnitude

physical quantity magnitude

Time is 12.76 s

Temperature is 36.8 C physical quantity magnitude

A scalar is a physical quantity thathas only a magnitude.

Examples:• Mass• Length• Time

• Temperature• Volume• Density

What is Vector?

California NorthCarolina

Position of California from North Carolina is 3600 km in west

physical quantity magnitudedirection

Displacement from USA to China is 11600 km in east

physical quantity magnitudedirection

USA China

A vector is a physical quantity that has both a magnitude and a direction. Examples:• Position• Displacement• Velocity

• Acceleration• Momentum• Force

Representation of a vector

Symbolically it is represented as

Representation of a vector

They are also represented by a single capitalletter with an arrow above it.

P⃗B⃗A⃗

Representation of a vector

Some vector quantities are represented by theirrespective symbols with an arrow above it.

F⃗v⃗r⃗

Position velocity Force

Types of Vectors

(on the basis of orientation)

Parallel Vectors

Two vectors are said to be parallel vectors, if they have same direction.

A⃗ P⃗

Q⃗B⃗

Equal VectorsTwo parallel vectors are said to be equal vectors,

if they have same magnitude.

A⃗B⃗

P⃗

Q⃗

A⃗=B⃗ P⃗=Q⃗

Anti-parallel Vectors

Two vectors are said to be anti-parallel vectors,if they are in opposite directions.

A⃗ P⃗

Q⃗B⃗

Negative Vectors

Two anti-parallel vectors are said to be negativevectors, if they have same magnitude.

A⃗B⃗

P⃗

Q⃗

A⃗=− B⃗ P⃗=− Q⃗

Collinear VectorsTwo vectors are said to be collinear vectors,

if they act along a same line.

A⃗

B⃗

P⃗

Q⃗

Co-initial VectorsTwo or more vectors are said to be co-initialvectors, if they have common initial point.

B⃗A⃗C⃗

D⃗

Co-terminus VectorsTwo or more vectors are said to be co-terminusvectors, if they have common terminal point.

B⃗A⃗

C⃗D⃗

Coplanar VectorsThree or more vectors are said to be coplanar

vectors, if they lie in the same plane.

A⃗

B⃗ D⃗

C⃗

Non-coplanar VectorsThree or more vectors are said to be non-coplanar

vectors, if they are distributed in space.

B⃗C⃗

A⃗

Types of Vectors

(on the basis of effect)

Polar Vectors

Vectors having straight line effect arecalled polar vectors.

Examples:

• Displacement• Velocity

• Acceleration• Force

Axial Vectors

Vectors having rotational effect arecalled axial vectors.

Examples:• Angular momentum• Angular velocity

• Angular acceleration• Torque

VectorAddition

(Geometrical Method)

Triangle Law

A⃗B⃗

A⃗B⃗

C⃗

= +

Parallelogram Law

A⃗ B⃗

A⃗ B⃗

A⃗ B⃗

B⃗ A⃗

C⃗

= +

Polygon Law

A⃗

B⃗

D⃗

C⃗

A⃗ B⃗

C⃗

D⃗

E⃗

= + + +

Commutative PropertyA⃗

B⃗B⃗

A⃗C⃗

C⃗

Therefore, addition of vectors obey commutative law.

= + = +

Associative Property

B⃗A⃗

C⃗

Therefore, addition of vectors obey associative law.

= + + = + +

D⃗ B⃗

A⃗

C⃗

D⃗

Subtraction of vectors

B⃗A⃗

−B⃗

A⃗

- The subtraction of from vector is defined as

the addition of vector to vector .

- = +

VectorAddition

(Analytical Method)

Magnitude of Resultant

P⃗O

B

A

CQ⃗ R⃗

M

θθ ⇒

OC2=OM 2+C M 2

OC2=(O A+AM )2+CM 2

OC2=OA2+2OA× AM+A C2

R2=P2+2 P×Q cos θ+Q2

R=√P2+2 PQ cosθ+Q2

Direction of Resultant

P⃗O A

CQ⃗ R⃗

Mα θ

⇒

⇒

B

Case I – Vectors are parallel ()P⃗ Q⃗ R⃗

R=√P2+2 PQ cos0 °+Q 2

R=√P2+2 PQ+Q2

R=√(P+Q)2

+¿ ¿

Magnitude: Direction:

R=P+Q

Case II – Vectors are perpendicular ()

P⃗ Q⃗ R⃗

R=√P2+2 PQ cos90 °+Q 2

R=√P2+0+Q2

+¿ ¿

Magnitude: Direction:

α=tan−1(QP )

P⃗

Q⃗

R=√P2+Q 2

α

Case III – Vectors are anti-parallel ()

P⃗ Q⃗ R⃗

R=√P2+2 PQ cos180 °+Q 2

R=√P2−2 PQ+Q2

R=√(P−Q)2

− ¿

Magnitude: Direction:

R=P−Q

If P>Q :

If P<Q :

Unit vectorsA unit vector is a vector that has a magnitude of

exactly 1 and drawn in the direction of given vector.

• It lacks both dimension and unit.• Its only purpose is to specify a direction in

space.

A⃗�̂�

• A given vector can be expressed as a product of its magnitude and a unit vector.

• For example may be represented as,

A⃗=A �̂�

Unit vectors

= magnitude of = unit vector along

Cartesian unit vectors

𝑥

𝑦

𝑧

�̂�

�̂�

�̂�

− 𝑦

−𝑥

−𝑧

-

-

-

Resolution of a VectorIt is the process of splitting a vector into two or more

vectors in such a way that their combined effect is same as that of

the given vector.

𝑡

𝑛

A⃗𝑡

A⃗𝑛A⃗

Rectangular Components of 2D Vectors

A⃗

A⃗𝑥

A⃗𝑦

O 𝑥

𝑦A⃗

A⃗=A 𝑥 �̂�+A 𝑦 �̂�θ

θA𝑥 �̂�

A𝑦 �̂�

A

A𝑥

A𝑦

θ

A𝑦=A sin θ

A𝑥=A cosθ

⇒

⇒

Rectangular Components of 2D Vectors

Magnitude & direction from components

A⃗=A 𝑥 �̂�+A 𝑦 �̂�

A=√ A𝑥2+A𝑦

2

Magnitude:

Direction:

θ= tan−1( A 𝑦

A𝑥)θ

A𝑥

A𝑦A

Rectangular Components of 3D Vectors

𝑥

𝑧

𝑦

A⃗A⃗𝑦

A⃗′A⃗𝑧

A⃗𝑥

A⃗=A⃗′+ A⃗ 𝑦

A⃗=A⃗ 𝑥+ A⃗ 𝑧+ A⃗ 𝑦

A⃗=A⃗ 𝑥+ A⃗ 𝑦+ A⃗𝑧

A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�

Rectangular Components of 3D Vectors

𝑥

𝑧

𝑦

𝛼

A

A𝑥

A𝑥=A cos𝛼

Rectangular Components of 3D Vectors

𝑥

𝑧

𝑦

AA𝑦𝛽

A𝑦=A cos 𝛽

Rectangular Components of 3D Vectors

𝑥

𝑧

𝑦

A

A𝑧

𝛾

A𝑧=A cos𝛾

Magnitude & direction from components

A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�

A=√ A𝑥2+A𝑦

2 +A𝑧2

Magnitude:

Direction:

𝛼=cos−1( A𝑥

A )𝛽=cos− 1( A 𝑦

A )𝛾=cos−1( A𝑧

A )

A

A𝑧

𝛾𝛽𝛼 A𝑥

A𝑦

Adding vectors by components

A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�B⃗=B𝑥 �̂�+B𝑦 �̂�+B𝑧 �̂�

Let us have

t hen

R𝑦=(A ¿¿ 𝑦+B𝑦)¿

Multiplyingvectors

Multiplying a vector by a scalar

• If we multiply a vector by a scalar s, we get a new vector.

• Its magnitude is the product of the magnitude of and the absolute value of s.

• Its direction is the direction of if s is positive but the opposite direction if s is negative.

Multiplying a vector by a scalarIf s is positive:

A⃗2 A⃗

If s is negative:A⃗

−3 A⃗

Multiplying a vector by a vector

• There are two ways to multiply a vector by a vector:

• The first way produces a scalar quantity and called as scalar product (dot product).

• The second way produces a vector quantity and called as vector product (cross product).

Scalar product

A⃗

B⃗θ

Examples of scalar product

W=F⃗ ∙ s⃗

W = work done F = force s = displacement

P=F⃗ ∙ v⃗

P = power F = force v = velocity

Geometrical meaning of Scalar dot product

A dot product can be regarded as the product of two quantities:

1.The magnitude of one of the vectors

2.The scalar component of the second vector along the direction of the first vector

Geometrical meaning of Scalar product

θ

AB cosθ

θ

A

A cos θBB

Properties of Scalar product

1The scalar product is commutative.

Properties of Scalar product

2The scalar product is distributive over

addition.

A⃗ ∙ (B⃗+C⃗ )=A⃗ ∙ B⃗+A⃗ ∙ C⃗

Properties of Scalar product

3The scalar product of two perpendicular

vectors is zero.

A⃗ ∙ B⃗=0

Properties of Scalar product

4The scalar product of two parallel vectors

is maximum positive.

A⃗ ∙ B⃗=A B

Properties of Scalar product

5The scalar product of two anti-parallel

vectors is maximum negative.

A⃗ ∙ B⃗=− A B

Properties of Scalar product

6The scalar product of a vector with itselfis equal to the square of its magnitude.

A⃗ ∙ A⃗=A2

Properties of Scalar product

7The scalar product of two same unit vectors is

one and two different unit vectors is zero.

�̂� ∙ �̂�= �̂� ∙ �̂�=�̂� ∙ �̂�=(1)(1)cos 0 °=1

�̂� ∙ �̂�= �̂� ∙ �̂�=�̂� ∙ �̂�=(1)(1)cos 90 °=0

Calculating scalar product using components

A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�B⃗=B𝑥 �̂�+B𝑦 �̂�+B𝑧 �̂�

Let us have

t hen

A⃗ ∙ B⃗=A𝑥B𝑥+A𝑦 B𝑦+A𝑧 B𝑧

Vector product

A⃗

B⃗θ

A⃗× B⃗=A B sin θ �̂�=C⃗

Right hand rule

A⃗

B⃗

C⃗

θ

Examples of vector product

τ⃗= r⃗× F⃗

= torque r = position F = force

L⃗=r⃗× p⃗L⃗=r p sin θ �̂�L = angular momentum r = position p = linear momentum

Geometrical meaning of Vector product

θ

A

B

B sin θ θ

A

B

Asinθ

= Area of parallelogram made by two vectors

Properties of Vector product

1The vector product is anti-commutative.

Properties of Vector product

2The vector product is distributive over

addition.

A⃗× ( B⃗+C⃗ )=A⃗× B⃗+A⃗×C⃗

Properties of Vector product

3The magnitude of the vector product of two

perpendicular vectors is maximum.

|A⃗× B⃗|=A B

Properties of Vector product

4The vector product of two parallel vectors

is a null vector.

A⃗× B⃗=0⃗

Properties of Vector product

5The vector product of two anti-parallel

vectors is a null vector.

A⃗× B⃗=0⃗

Properties of Vector product

6The vector product of a vector with itself

is a null vector.

A⃗× A⃗=0⃗

Properties of Vector product

7The vector product of two same unit

vectors is a null vector.

�̂�× �̂�= �̂�× �̂�=�̂�×�̂�¿ (1)(1)sin 0 ° �̂�=0⃗

Properties of Vector product

8The vector product of two different unit

vectors is a third unit vector.�̂�× �̂�=�̂��̂�× �̂�=�̂��̂�× �̂�= �̂�

�̂�× �̂�=− �̂��̂�× �̂�=− �̂��̂�×�̂�=− �̂�

Aid to memory

Calculating vector product using components

A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�B⃗=B𝑥 �̂�+B𝑦 �̂�+B𝑧 �̂�

Let us have

t hen

Calculating vector product using components

A⃗× B⃗=| �̂� �̂� �̂�A𝑥 A 𝑦 A𝑧

B𝑥 B𝑦 B𝑧|

Thankyou