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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
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