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
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