Vector Algebra

SYNOPSIS

Vector Algebra

1. 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.
2. Magnitude of a vector 
3.  Distance between 2 points in 3D plane is given by

   d = |PQ| =
   
   

4. 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 =

5. Types of Vectors:
   

   Zero Vector: A vector having zero magnitude, i.e. if =  0. Also, it has no direction. Co-initial 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.

6. 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.

7. Linear combination/dependence/independence
   
   
   1. Linear Combination:
      A vector  is said to be a linear combination of vectors  if there exist scalars such that
        .
   2. Linearly Independent:
      A system of vectors  is said to be linearly independent if for  such that 
       
      
      
   3. Linearly Dependent:
      A system of vectors 
       are said to be linearly dependent if there exist scalars 
      not all zero, such that
      
   
   
8. Vector Lines:
   To determine vector equation of a line, we need
   i. A point on the line
   ii. A vector parallel to the line
   
   
9. Vector Planes:To determine vector equation of a plane, we need
   1. A point on the plane
   2. A vector perpendicular to the line
   
   
   

10. Scalar or dot product of vectors:
    Scalar product of vectors and  is the projection of  over  .
    Denoted by . and given by 
    . = 
    
    

11. Vector or cross product of vectors:
    Vector product of vectors  and  is written as
    and it is defined as
     sin⁡θ , where n ̂ is a unit vector
    along the line perpendicular to both  and .
    
    

12. 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.
    

13. Vector triple product:
    Vector triple product of three vectors  and  is the vector