VECTOR ALGEBRA
Wednesday, 25 October, 2023 1:40 PM
SCALARS & VECTOR QUANTITY/FIELD
UNIT VECTOR
VECTOR OPERATION
- ADDITION, SUBTRACTION
- POSITION, DISTANCE VECTORS
- VECTOR MULTIPLICATION
○ DOT PRODUCT
○ CROSS PRODUCT
- COMPONENTS OF A VECTOR
SCALARS & VECTOR QUANTITY/FIELD
Scalar quantity - physical quantity that only describes magnitude
Eg: time, mass, distance, temperature, energy etc
Vector quantity - physical quantity that describes both magnitude and direction
Eg: velocity (speed with direction), force (magnitude with direction)
Nomenclature of Scalar and vector quantity used in this module:
Scalar unknown - A, B
Vector unknown - , (watch out for the arrow above the unknown)
EM theory deals with study/behaviour of Electric field, Magnetic field and their relationship
Field - a function that specifies a particular quantity everywhere in a region (2D / 3D space)
- indicates variation of a quantity (scalar/vector) as a function of space (locations) and/or time
Scalar field - temperature distribution in a room, sound intensity in a cinema, electric potential in a region, topography
Vector field - velocity of raindrops/wind around the atmosphere, gravitational/electric/magnetic field
CHP1-Vector Analysis Notes Page 1
, UNIT VECTOR
A vector quantity in cartesian coordinates (xy 2D or xyz 3D) can be represented as
(2D)
(3D)
, , - unit vector along x, y, z axis, whose magnitude is unity.
Magnitude of
In general, unit vector of :
VECTOR OPERATION
Addition / Subtraction:
CHP1-Vector Analysis Notes Page 2
,Position Vectors, Distance Vectors:
A POINT P in Cartesian Coordinate is represented by (x, y, z)
Position vector of POINT P = directed distance from origin O (0,0,0) to P
Example:
Distance vector = displacement from one point to another.
If position vector and
Then the displacement from P to Q:
Take note on the confusion between point P, its position vector and a vector field
Vector Multiplication:
Dot product of two vectors and
Where is the angle between and
If and , then
gives a scalar quantity.
The quantity is a measure of how much the components of goes in the direction of / how closely two vectors align / Projection
of on .
If , means two vectors are orthogonal, perpendicularly to each other.
If , means two vectors are parallel to each other.
Cross product of two vectors and
CHP1-Vector Analysis Notes Page 3
, Cross product of two vectors and
Where is the angle formed from to , is the unit vector normal to the plane containing and
If and , then
gives a vector quantity.
a measure of how much two vectors point in different directions.
is a vector perpendicular to both and .
Magnitude of = area of parallelogram formed by and
follows right hand rule:
COMPONENTS OF A VECTOR
Projection/Components OF Vector on Vector
How much is the Vector projected on Vector
Components can be scalar or vector.
CHP1-Vector Analysis Notes Page 4
Wednesday, 25 October, 2023 1:40 PM
SCALARS & VECTOR QUANTITY/FIELD
UNIT VECTOR
VECTOR OPERATION
- ADDITION, SUBTRACTION
- POSITION, DISTANCE VECTORS
- VECTOR MULTIPLICATION
○ DOT PRODUCT
○ CROSS PRODUCT
- COMPONENTS OF A VECTOR
SCALARS & VECTOR QUANTITY/FIELD
Scalar quantity - physical quantity that only describes magnitude
Eg: time, mass, distance, temperature, energy etc
Vector quantity - physical quantity that describes both magnitude and direction
Eg: velocity (speed with direction), force (magnitude with direction)
Nomenclature of Scalar and vector quantity used in this module:
Scalar unknown - A, B
Vector unknown - , (watch out for the arrow above the unknown)
EM theory deals with study/behaviour of Electric field, Magnetic field and their relationship
Field - a function that specifies a particular quantity everywhere in a region (2D / 3D space)
- indicates variation of a quantity (scalar/vector) as a function of space (locations) and/or time
Scalar field - temperature distribution in a room, sound intensity in a cinema, electric potential in a region, topography
Vector field - velocity of raindrops/wind around the atmosphere, gravitational/electric/magnetic field
CHP1-Vector Analysis Notes Page 1
, UNIT VECTOR
A vector quantity in cartesian coordinates (xy 2D or xyz 3D) can be represented as
(2D)
(3D)
, , - unit vector along x, y, z axis, whose magnitude is unity.
Magnitude of
In general, unit vector of :
VECTOR OPERATION
Addition / Subtraction:
CHP1-Vector Analysis Notes Page 2
,Position Vectors, Distance Vectors:
A POINT P in Cartesian Coordinate is represented by (x, y, z)
Position vector of POINT P = directed distance from origin O (0,0,0) to P
Example:
Distance vector = displacement from one point to another.
If position vector and
Then the displacement from P to Q:
Take note on the confusion between point P, its position vector and a vector field
Vector Multiplication:
Dot product of two vectors and
Where is the angle between and
If and , then
gives a scalar quantity.
The quantity is a measure of how much the components of goes in the direction of / how closely two vectors align / Projection
of on .
If , means two vectors are orthogonal, perpendicularly to each other.
If , means two vectors are parallel to each other.
Cross product of two vectors and
CHP1-Vector Analysis Notes Page 3
, Cross product of two vectors and
Where is the angle formed from to , is the unit vector normal to the plane containing and
If and , then
gives a vector quantity.
a measure of how much two vectors point in different directions.
is a vector perpendicular to both and .
Magnitude of = area of parallelogram formed by and
follows right hand rule:
COMPONENTS OF A VECTOR
Projection/Components OF Vector on Vector
How much is the Vector projected on Vector
Components can be scalar or vector.
CHP1-Vector Analysis Notes Page 4
