•
•
•
•
•
•
•
•
•
•
Recaps
Vector Addition (different cases)
Unit Vectors
Distributive law for vector addition
3D Vectors
Direction cosines (details)
Multiplication with a scalar
Scalar or Dot Product
Vector or Cross Product
Conclusion
Vector Addition Case I – Vectors are parallel (θ = 0°)
P
Q
+
Magnitude:
R
=
Direction:
Vector Addition Case II – Vectors are perpendicular (θ=90°)
P
+
Magnitude:
R
Q
α
=
Direction:
Vector Addition Case I – Vectors are anti-parallel (θ = 180°)
P
Q
Magnitude:
R
=
Direction:
Unit vectors
• A given vector can be expressed as a product of its
magnitude and a unit vector.
• For example A may be represented as,
Distributive Law of Vector Addition
M (A + B) = m A + m B
mB
B
mA + mB
A+B
A
mA
3D Vectors :Rectangular Components
A  Ax  Az
3D Vectors: Rectangular Components
3D Vectors: Rectangular Components
3D Vectors: Rectangular Components
Final Magnitude and Direction Cosines
Magnitude:
Direction Cosines:
Multiplication with a scalar
• If we multiply a vector A by a scalar s, we get a new vector.
• Its magnitude is the product of the magnitude of A and the absolute
value of s.
• Its direction is the direction of A if s is positive but the opposite
direction if s is negative.
If s is positive:
If s is negative:
A
A
2A
-2 A
Multiplying
Vector with A Vector
Multiplying Vector with 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 or Dot Product
B
θ
A
Scalar Product or Dot Product
Examples
W = work done
F = force
s = displacement
P = power
F = force
v = velocity
Scalar Product or Dot Product
Geometrical Meaning
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
Scalar Product or Dot Product
Properties of Scalar product
1. The scalar product is commutative.
2. The scalar product is distributive over addition.
3. The scalar product of two perpendicular vectors is zero.
Properties of Scalar product
4. The scalar product of two parallel vectors is maximum positive.
5. The scalar product of two anti-parallel vectors is maximum negative.
6. The scalar product of a vector with itself is equal to the square of its
magnitude.
Properties of Scalar product
7. The scalar product of two same unit vectors is one and two different
unit vectors is zero.
Let us have
then
Properties of Scalar product
Hence
Vector Product or Cross Product
B
θ
A
Vector Product or Cross Product
Right hand rule
C
B
n̂
θ
A
Vector Product or Cross Product
Examples
τ = torque
r = position
F = force
L = angular momentum
r = position
p = linear momentum
Vector Product or Cross Product
Geometrical meaning of Vector product
Properties of Vector Product
1. The vector product is anti-commutative.
Hence
2. The vector product is distributive over addition
Properties of Vector Product
3. The magnitude of the vector product of two perpendicular vectors is
maximum.
4. The vector product of two parallel vectors is a null vector.
5. The vector product of two anti-parallel vectors is a null vector.

Properties of Vector Product
6. The vector product of a vector with itself is a null vector.

7. The vector product of two same unit vectors is a null vector.
8. The vector product of two different unit vectors is a third unit vector.
and
Properties of Vector Product
and
Vector Product using Components
Given two vectors
Their Vector product goes as -
Vector Product using Components
In these two lectures
• We started with definition of scalars and vectors
• We defined different types of vectors
•We defined components of 2D and 3D vectors and their operations
• We defined addition and subtraction of vectors
•We defined scalar and vector multiplications
•Finally we explained the properties of Addition and multiplication of vectors