Class 12 Maths Revision Notes for Vector Algebra of Chapter 10

CBSE Class 12 Mathematics Revision Notes Chapter 10 Vector Algebra

Vector: A quantity that has magnitude as well as direction is called vector.

• Zero Vector: A vector whose intial and terminal point coincide is called a zero vector or a null vector. It is denoted as .
• Co-initial vectors: Two or more vectors having the same initial points are called co-initial vectors.
• Collinear vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
• Equal vectors: Two vectors are said to be equal, if they have the same magnitude and direction regardless of the position of their initial points.
• Negative of a vector: A vector whose magnitude is the same as that of a given vector, but direction is opposite to that of it, is called negative of the given vector.
• Position vector of a point P (x, y) is given as   and its magnitude by 
• The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.
• The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as: 
• The vector sum of the three sides of a triangle taken in order is  
• The vector sum of two conidial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
• The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) according as the value of λ is positive (or negative).
• For a given vector , the vector gives the unit vector in the direction of 
• The position vector of a point R dividing a line segment joining the points P and Q whose position vectors are   and respectively, in the ratio 
  (i)  internally, is given by 
  (ii)  externally, is given by 
• The scalar product of two given vectors  and  having angle θ between them is defined as 
  Also, when is given, the angle between the vectors  and  may be determined by 
• If  is the angle between two vector   and ,  then their cross product is given as
  whereis a unit vector perpendicular to the plane containing  and . Such that  form right handed system of coordinate axes.
• If we have two vectors   and  given in component form as   and    and  be  any scalar, then,
  
  
  
  
  Parallelogram Law of vector addition: If two vectors  and  are represented by adjacent sides of a parallelogram in magnitude and direction, then their sum  is represented in magnitude and direction by the diagonal of the parallelogram through their common initial point. This is known as Parallelogram Law of vector addition.