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JEE Maths Vector Algebra

Vector Algebra PDF Notes, Important Questions and Synopsis

SYNOPSIS

Vector Algebra

  1. Vector is a quantity having both magnitude and direction.
    Note: A directed line segment is a vector denoted by begin mathsize 12px style AB with rightwards arrow on top end style or simply begin mathsize 12px style straight a with rightwards arrow on top equals straight a subscript straight x straight i with hat on top plus straight a subscript straight y straight j with hat on top plus straight a subscript straight z straight k with hat on top end style
    where begin mathsize 12px style straight a subscript straight x comma straight a subscript straight y comma straight a subscript straight z end style denotes ith, jth, kth components.
  2. Magnitude of a vector begin mathsize 12px style straight a with rightwards arrow on top equals straight a subscript straight x straight i with hat on top plus straight a subscript straight y straight j with hat on top plus straight a subscript straight z straight k with hat on top text  is  end text open vertical bar straight a with rightwards arrow on top close vertical bar equals square root of left parenthesis straight a subscript straight x right parenthesis squared plus left parenthesis straight a subscript straight y right parenthesis squared plus left parenthesis straight a subscript straight z right parenthesis squared end root end style
  3.  Distance between 2 points in 3D plane is given by

    d = |PQ| =begin mathsize 12px style square root of open parentheses straight x subscript 2 minus straight x subscript 1 close parentheses squared plus open parentheses straight y subscript 2 minus straight y subscript 1 close parentheses squared plus open parentheses straight z subscript 2 minus straight z subscript 1 close parentheses squared end root end style

  4. Internal/External division:

    •  Internal division
      Let P and Q be the two given points. Let begin mathsize 12px style straight R left parenthesis straight r with rightwards arrow on top right parenthesis end style  be the point which divides PQ internally in the ratio m : n.


      Then its coordinates are
      R = begin mathsize 12px style fraction numerator mx subscript 2 plus nx subscript 1 over denominator straight m plus straight n end fraction comma fraction numerator my subscript 2 plus ny subscript 1 over denominator straight m plus straight n end fraction comma fraction numerator mz subscript 2 plus nz subscript 1 over denominator straight m plus straight n end fraction end style
     
    • External Division
      Let P and Q be the two given points. Let  begin mathsize 12px style straight R left parenthesis straight r with rightwards arrow on top right parenthesis end style  be the point which divides PQ internally in the ratio m : n.


      Then its coordinates are
      R =begin mathsize 12px style fraction numerator mx subscript 2 minus nx subscript 1 over denominator straight m minus straight n end fraction comma fraction numerator my subscript 2 minus ny subscript 1 over denominator straight m minus straight n end fraction comma fraction numerator mz subscript 2 minus nz subscript 1 over denominator straight m minus straight n end fraction end style
  5. Types of Vectors:
    •  Zero Vector:
      A vector having zero magnitude, i.e.
      if = begin mathsize 12px style vertical line straight a with rightwards arrow on top vertical line end style 0. Also, it has no direction.
    •   Co-initial vectors:
      Two or more vectors having the same initial point.
    •   Collinear Vectors:
      Vectorsbegin mathsize 12px style straight a with rightwards arrow on top end style and begin mathsize 12px style straight b with rightwards arrow on top end styleare 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  
      begin mathsize 12px style vertical line straight a with rightwards arrow on top vertical line end style= 1. It is denoted by begin mathsize 12px style straight a with hat on top equals fraction numerator straight a with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar end fraction end style  
      .
    • Equal vectors:
      Vectors begin mathsize 12px style straight a with rightwards arrow on top end style and  begin mathsize 12px style straight b with rightwards arrow on top end style are equal if  & 
      begin mathsize 12px style straight a with hat on top end stylebegin mathsize 12px style straight b with hat on top end style .
    • 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 begin mathsize 12px style OP with rightwards arrow on top end style 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
        begin mathsize 12px style AC with rightwards arrow on top equals AB with rightwards arrow on top plus BC with rightwards arrow on top end style.
       This is known as the triangle law of vector addition.
        Also, if we have begin mathsize 12px style AB with rightwards arrow on top equals straight a with rightwards arrow on top end style & begin mathsize 12px style BC with rightwards arrow on top equals straight b with rightwards arrow on top end style , then
          begin mathsize 12px style AC with rightwards arrow on top equals straight a with rightwards arrow on top plus straight b with rightwards arrow on top end style
     

     ii. Multiplication of a vector by a scalar:
        Let  begin mathsize 12px style straight a with rightwards arrow on top end style be the vector and k be a scalar. 
         Product of  begin mathsize 12px style straight a with rightwards arrow on top end style and k is begin mathsize 12px style straight k straight a with rightwards arrow on top end style , where each  component          of  begin mathsize 12px style straight a with rightwards arrow on top end styleis multiplied by k.
     

  7. Linear combination/dependence/independence

    1. Linear Combination:
      A vector begin mathsize 10px style straight a with rightwards arrow on top end style is said to be a linear combination of vectors begin mathsize 12px style stack straight a subscript 1 with rightwards arrow on top comma stack straight a subscript 2 with rightwards arrow on top comma stack straight a subscript 3 with rightwards arrow on top comma. .. comma stack straight a subscript straight n with rightwards arrow on top comma end style if there exist scalars such that
       begin mathsize 12px style straight a with rightwards arrow on top equals straight m subscript 1 stack straight a subscript 1 with rightwards arrow on top plus straight m subscript 2 stack straight a subscript 2 with rightwards arrow on top plus. .. plus straight m subscript straight n stack straight a subscript straight n with rightwards arrow on top end style .
    2. Linearly Independent:
      A system of vectors begin mathsize 12px style stack straight a subscript 1 with rightwards arrow on top comma stack straight a subscript 2 with rightwards arrow on top comma stack straight a subscript 3 with rightwards arrow on top comma. .. comma stack straight a subscript straight n with rightwards arrow on top end style is said to be linearly independent if fobegin mathsize 12px style straight m subscript 1 comma straight m subscript 2 comma straight m subscript 3 comma.. comma straight m subscript straight n end style such that begin mathsize 12px style straight m subscript 1 stack straight a subscript 1 with rightwards arrow on top plus straight m subscript 2 stack straight a subscript 2 with rightwards arrow on top plus. .. plus straight m subscript straight n stack straight a subscript straight n with rightwards arrow on top equals 0 with rightwards arrow on top end style
       begin mathsize 12px style straight m subscript 1 equals straight m subscript 2 equals. ... equals straight m subscript straight n equals 0. end style

    3. Linearly Dependent:
      A system of vectors 
      begin mathsize 12px style stack straight a subscript 1 with rightwards arrow on top comma stack straight a subscript 2 with rightwards arrow on top comma stack straight a subscript 3 with rightwards arrow on top comma. .. comma stack straight a subscript straight n with rightwards arrow on top end style are said to be linearly dependent if there exist scalars begin mathsize 12px style straight m subscript 1 comma straight m subscript 2 comma straight m subscript 3 comma.. comma straight m subscript straight n comma end style
      not all zero, such that
      begin mathsize 12px style straight m subscript 1 stack straight a subscript 1 with rightwards arrow on top plus straight m subscript 2 stack straight a subscript 2 with rightwards arrow on top plus. .. plus straight m subscript straight n stack straight a subscript straight n with rightwards arrow on top equals 0 with rightwards arrow on top. end style

  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 vectorsbegin mathsize 12px style straight a with rightwards arrow on top end style and begin mathsize 12px style b with rightwards arrow on top end style is the projection of begin mathsize 12px style straight a with rightwards arrow on top end style over begin mathsize 12px style b with rightwards arrow on top end style .
    Denoted by begin mathsize 12px style straight a with rightwards arrow on top end style.begin mathsize 12px style b with rightwards arrow on top end style and given by 
    begin mathsize 12px style straight a with rightwards arrow on top end stylebegin mathsize 12px style b with rightwards arrow on top end stylebegin mathsize 12px style vertical line straight a with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line cosθ end style

  11. Vector or cross product of vectors:
    Vector product of vectors begin mathsize 12px style straight a with rightwards arrow on top end style and begin mathsize 12px style b with rightwards arrow on top end style is written as
    begin mathsize 12px style straight a with rightwards arrow on top cross times straight b with rightwards arrow on top end styleand it is defined as
    begin mathsize 12px style straight a with rightwards arrow on top cross times straight b with rightwards arrow on top equals open vertical bar straight a with rightwards arrow on top close vertical bar open vertical bar straight b with rightwards arrow on top close vertical bar sinθ space straight n with hat on top end style sin⁡θ begin mathsize 12px style straight n with hat on top end style, where n ̂ is a unit vector
    along the line perpendicular to both begin mathsize 12px style straight a with rightwards arrow on top end style and begin mathsize 12px style b with rightwards arrow on top end style.

  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 begin mathsize 12px style straight a with rightwards arrow on top comma straight b with rightwards arrow on top end style and begin mathsize 12px style straight c with rightwards arrow on top end style is begin mathsize 12px style straight a with rightwards arrow on top times left parenthesis straight b with rightwards arrow on top cross times straight c with rightwards arrow on top right parenthesis end style .
    It represents the volume of the parallelepiped.
    Also, the volume of a tetrahedron isbegin mathsize 12px style 1 over 6 end style th times the volume of the parallelepiped.

  13. Vector triple product:
    Vector triple product of three vectors begin mathsize 12px style straight a with rightwards arrow on top comma straight b with rightwards arrow on top end style and begin mathsize 12px style straight c with rightwards arrow on top end style is the vector begin mathsize 12px style straight a with rightwards arrow on top cross times open parentheses straight b with rightwards arrow on top cross times straight c with rightwards arrow on top close parentheses end style