Overview
Classification of Vectors
» Classification by magnitude
→ null or zero vector : magnitude
→ unit vector : magnitude
→ proper vector : magnitude not
» By similarities of two vectors
→ equal vectors : All corresponding components equal
→ like vectors: same directions
→ unlike vectors : different directions
→ co-initial vectors : same initial point
→ co-linear vectors : on the same line
→ co-planar vectors : on the same plane
→ non-co-planar vectors : not on the same plane
null
The word 'null' means 'zero; nothing'.
A vector with zero magnitude is a 'zero or null vector'.
All the following vectors have magnitude
A null vector is a scalar as well as a vector.
A null vector is also called improper vector as it does not have a direction.
• Zero or Null Vector (magnitude )
Null Vector or Zero Vector: A quantity of zero magnitude, given as or . For calculations, it can be used as .
Technically a null vector is not a vector. Arithmetic operations on vectors like addition, may result in a null vector.
can be called any one of the following:
null vector
improper vector
zero vector
scalar
proper
The word 'proper' means 'correct type or form'.
'A vector with direction' is a 'proper vector'.
Proper Vector: A vector with non-zero magnitude.
If vector is a proper vector then
unit
The word 'unit' means
'A vector having magnitude ' is an 'unit vector'.
Unit Vector: A vector with magnitude 1.
If vector is an unit vector, then
Is an unit vector?
The magnitude is not
The magnitude is
So the given vector is not a unit vactor.
equal
The word 'Equal' means 'being same in quantity or value'.
Two vectors are 'equal' when both the magnitudes and directions are same.
Equal Vectors: The two vectors and are Equal
if and only if
Two vectors and are equal, then the following are true
directional cosines of and are equal
like & unlike
The word 'like' means 'having same characteristics or properties'.
Two vectors called 'like vectors' 'when the vectors have same direction'.
Like Vectors: Vectors of same direction.
The vectors and are like vectors, if
Unlike Vectors: Vectors of different direction.
The vectors and are unlike vectors, if
Two vectors and are like vectors, then the following are true
directional cosines of and are equal
angles made with , , axes of are equal to that of .
initial
The word 'initial' means 'beginning or starting'.
The prefix 'co' in co-initial means 'jointly; mutually'.
Two vectors are 'co-initial' vectors 'When the vectors start from the same position'.
A vector can be positioned at any point without modifying the defining parameters magnitude and direction. When vectors are used to represent shapes or quantities, the position of the vector is additionally specified.
Co-initial Vectors:Two vectors and are co-initial vectors when they are positioned at the same starting point .
Given two vectors, and , are these vectors co-initial?
The answer is 'Cannot determine from the given information'. The initial position of the vector is to be given separately and when not given, the vectors can be positioned anywhere.
collinear
The word 'collinear' means 'of lying in the same line'. "co" means "together; jointly" ; and "linear" means "line".
Two vectors are called 'collinear' vectors 'When the vectors are on the same line'.
Collinear Vectors: Two vectors and are collinear vectors if where .
Given that two vectors are collinear, does that also imply that the given vectors are like vectors?
'No.'
Collinear vectors can be either in same direction or in opposite direction.
Furthermore, there are two parameters to note in collinear vectors when comparing them for being like vectors.
1. the direction - whether they are in same direction or in the opposite direction.
2. the position - vectors may be positioned at different points. Like vectors having same direction may not be collinear because of the position.
co-planar
The word 'co-planar' means 'of lying in the same plane'.
Two vectors are 'coplanar' 'When two vectors are in the same plane'
Co-planar Vectors: Two vectors and are coplanar if they lie on the same plane.
Under condition that the positions of vectors are not specified, and the vectors can be equivalently placed anywhere in the 3-D space, any two vectors will be coplanar.
Three vectors , , are co-planar (under the condition that vectors are equivalently positioned anywhere in the 3-D space), if
Co-planar property in terms of vector product is given as .
Non-co-planar Vectors: Three vectors , , are non-co-planar (under the condition that vectors are equivalently positioned anywhere in the 3-D space), if
Co-planar property in terms of vector product is given as .
Are and coplanar?
The answer is 'Yes, they lie in the xy-plane'.
negative
The word 'negative' means 'opposite; reverse'
The negative of is
Negative of a Vectors: For the vector , the negative of is
What is the 'negative' of vector ?
The answer is ''.
component
The word 'component' means 'constituent part of a larger whole'.
The components of a vector are 'x, y, and z components along the three axes'.
Component Form of a Vector: A vector is given in the component form as
,
where are the components along -axes respectively.
What is the component form of unit vector along -axis?
The answer is '', in the usual convention of representing component along x-axis using .
summary
Classification of Vectors
» Classification by magnitude
→ null or zero vector : magnitude
→ unit vector : magnitude
→ proper vector : magnitude not <
» By similarities of two vectors
→ equal vectors : All corresponding components equal
→ like vectors: same directions
→ unlike vectors : different directions
→ co-initial vectors : same initial point
→ co-linear vectors : on the same line
→ co-planar vectors : on the same plane
→ non-co-planar vectors : not on the same plane
Outline
The outline of material to learn vector-algebra is as follows.
Note: Click here for detailed outline of vector-algebra.
• Introduction to Vectors
→ Introducing Vectors
→ Representation of Vectors
• Basic Properties of Vectors
→ Magnitude of Vectors
→ Types of Vectors
→ Properties of Magnitude
• Vectors & Coordinate Geometry
→ Vectors & Coordinate Geometry
→ Position Vector of a point
→ Directional Cosine
• Role of Direction in Vector Arithmetics
→ Vector Arithmetics
→ Understanding Direction of Vectors
• Vector Addition
→ Vector Additin : First Principles
→ Vector Addition : Component Form
→ Triangular Law
→ Parallelogram Law
• Multiplication of Vector by Scalar
→ Scalar Multiplication
→ Standard Unit Vectors
→ Vector as Sum of Vectors
→ Vector Component Form
• Vector Dot Product
→ Introduction to Vector Multiplication
→ Cause-Effect-Relation
→ Dot Product : First Principles
→ Dot Product : Projection Form
→ Dot Product : Component Form
→ Dot Product With Direction
• Vector Cross Product
→ Vector Multiplication : Cross Product
→ Cross Product : First Principles
→ Cross Product : Area of Parallelogram
→ Cross Product : Component Form
→ Cross Product : Direction Removed