Vector Algebra
Vector Algebra PDF Notes, Important Questions and Synopsis
SYNOPSIS
Vector Algebra
 Vector is a quantity having both magnitude and direction.
Note: A directed line segment is a vector denoted by or simply
where denotes i^{th}, j^{th}, k^{th} components.  Magnitude of a vector
 Distance between 2 points in 3D plane is given by
d = PQ =

Internal/External division:
 Internal division
Let P and Q be the two given points. Let be the point which divides PQ internally in the ratio m : n.
Then its coordinates are
R =
 External Division
Let P and Q be the two given points. Let be the point which divides PQ internally in the ratio m : n.
Then its coordinates are
R =
 Internal division
 Types of Vectors:
 Zero Vector:
A vector having zero magnitude, i.e.
if = 0. Also, it has no direction.  Coinitial vectors:
Two or more vectors having the same initial point.  Collinear Vectors:
Vectors and are said to be collinear if they are parallel to each other.  Free vectors:
Vectors whose initial points are not specified.
 Unit Vector:
A vector whose magnitude is 1, i.e. if
= 1. It is denoted by .  Equal vectors:
Vectors and are equal if & = .  Coplanar vectors:
Vectors which are parallel or lying in the same plane are coplanar.  Localised vectors:
Vectors drawn parallel to a given vector, but through a specified point as the initial point.
Position vector: A vector having O and P as its initial and terminal points, is called the position vector of point P, where O is the origin.  Zero Vector:
 Operations on vectors:
i. Addition of vectors:
A, B and C are three points, then
.
This is known as the triangle law of vector addition.
Also, if we have & , then
ii. Multiplication of a vector by a scalar:Let be the vector and k be a scalar.Product of and k is , where each component of is multiplied by k.  Linear combination/dependence/independence
 Linear Combination:
A vector is said to be a linear combination of vectors_{ } if there exist scalars such that
.  Linearly Independent:
A system of vectors _{} is said to be linearly independent if for _{} such that
 Linearly Dependent:
A system of vectors
_{} are said to be linearly dependent if there exist scalars _{}
not all zero, such that
 Linear Combination:
 Vector Lines:
To determine vector equation of a line, we need
i. A point on the line
ii. A vector parallel to the line  Vector Planes:To determine vector equation of a plane, we need
 A point on the plane
 A vector perpendicular to the line

Scalar or dot product of vectors:
Scalar product of vectors and is the projection of over .
Denoted by . and given by
. = 
Vector or cross product of vectors:
Vector product of vectors and is written as
and it is defined as
sinθ , where n ̂ is a unit vector
along the line perpendicular to both and . 
Scalar triple product:
The dot product of one of the vectors with the cross product of the other two.
i.e. Scalar triple product of three vectors and is .
It represents the volume of the parallelepiped.
Also, the volume of a tetrahedron is th times the volume of the parallelepiped. 
Vector triple product:
Vector triple product of three vectors and is the vector
Related Chapters
 Sets, Relations and Functions
 Complex Numbers and Quadratic Equations
 Matrices and Determinants
 Permutations and Combinations
 Mathematical Induction
 Binomial Theorem and its Simple Applications
 Sequences and Series
 Limit, Continuity and Differentiability
 Integral Calculus
 Differential Equations
 Coordinate Geometry
 Three Dimensional Geometry
 Statistics and Probability
 Trigonometry
 Mathematical Reasoning