Important Formulas: Vector - Mathematics (Maths) Class 12 - JEE

1. Definitions:
A Vector may be described as a quantity having both magnitude & direction. A vector is generally represented by a directed line segment, say   . A is called the initial point & B is called the terminal point. The magnitude of vector  is expressed by.

Zero vector  a vector of zero magnitude i.e.which has the same initial & terminal point, is called a Zero vector. It is denoted by O.

Unit vector a vector of unit magnitude in direction of a vector  is called unit vector along  and is denoted by aˆ symbolically

Equal vectors  two vectors are said to be equal if they have the same magnitude, direction & represent the same physical quantity.  

Collinear vectors  two vectors are said to be collinear if their  directed line segments are parallel disregards to their direction. Collinear vectors are also called Parallel vectors. If they have the same direction they are named as like vectors otherwise.

unlike vectors. Symbolically,  two  non  zero  vectors  are collinear if and only if, 

where K ∈ R  Coplanar vectors a given number of vectors are called coplanar if their line segments are all parallel to the same plane. Note that  “two vectors are always coplanar”. position vector  let O be a fixed origin, then the position vector of a point P is the vector →OP . If  a & b & position vectors of two point A and B, then  , →AB = b a

− =  pv of B −  pv of A  .

2. Vector addition: 

If two vectors  are represented by →  , then their sum  is a vector represented by  , where OC is the diagonal  of the parallelogram OACB.

 (commutative)

  (associativity)

3. Multiplication of vector by scalars :

If  is a vector & m is a scalar, then m is a vector parallel to whose modulus is |m| times that  of . This multiplication is called Scalar multiplication. If are vectors & m, n are scalars, then:

4. Section formula :

If   are the position vectors of two points A & B then the p.v. of a point which divides AB in the ratio m : n is given by :   

note p.v..of mid point of AB =

5. Direction Cosines 

Let   the angles which this  vector makes  with the +ve directions OX,OY & OZ are called Direction angles & their cosines are called the Direction cosines   Note that, cos²    αααα + cos² ββββ + cos² ΓΓΓΓ = 1.

6. Vector equation of a line:
Parametric vector  equation of a line passing through two point

 is given by, where t is a parameter. If the line passes  through  the  point & is parallel to the vector then   its equation is,

Note that the equations of the bisectors of the angles between the lines  

7. Test of collinearity  :
Three points A,B,C with position vectors

 respectively are collinear,  if & only if there exist scalars x , y , z  not all zero simultaneously such that  ; where x + y + z = 0.

8. Scalar product of two vectors:

 note that if θ is acute then  &  if θ is obtuse  then

 (commutative)

Note: That vector component of  and perpendicular to  the angle φ between  is given by  cos φ =  0 ≤ φ ≤ π.

Note : (i) Maximum  value  of  

(ii) Minimum  values  of  

(iii) Any  vector can  be  written  as  

(iv) A vector in the direction of the bisector of the angle between the two vectors   Hence bisector  of  the  angle  between the two vector  where  Bisector of the exterior angle between

9. Vector product of two vectors :

(i) If  are two vectors   &   θ   is   the  angle  between  them  then  where  is the unit vector perpendicular to both   such  that      forms a  right  handed  screw  system.

(ii) Lagranges Identity : for any two vectors

(iii) Formulation of vector product in terms of scalar product:
The vector product  is the vector   such that.

(i)  form a right handed system.

(iv)  are parallel (collinear)  where K is a scalar..

(vi) Geometrically  of  the  parallelogram  whose  two  adjacent  sides are represented by

(vii) Unit vector perpendicular to the plane of

- A vector of magnitude ‘r ’ & perpendicular to the palne of  

If θ is the angle between

(viii) Vector area  If  are the pv’s of 3 points  A, B & C then the vector area of triangle ABC  The points A, B & C are collinear if  

Area of  any quadrilateral whose diagonal vectors are

10. Shortest distance between two lines:
If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & are also not parallel are called SKEW LINES. For Skew lines the direction of the shortest distance would be perpendicular to both the lines. The magnitude of the shortest distance vector would be equal to that of the projection of  along the direction of the line of shortest distance,  is parallel to   i. e .     

1. The  two  lines  directed  along  will  intersect  only  if  shortest distance = 0  i.e. 

 lies in the plane containing

2. If two lines are given by  i.e.  they are parallel then

11. Scalar triple product / box product / mixed product :
The  scalar  triple  product  of  three  vectors  is  defined  as :

 sin θ cos φ where θ is the angle between  it is also defined as  spelled as box product .Scalar triple product geometrically represents the volume of the parallelopiped  whose three  couterminous edges are represented by  

In a scalar triple product the position of dot & cross can be interchanged i.e.

 where  are non coplanar vectors .

If  are coplanar  

Scalar product of three vectors, two of which are equal or parallel is 0 i.e.

Note : If  are  non − coplanar  then   for  right  handed  system  &  for left handed system .

The volume of the tetrahedron OABC with O as origin & the pv’s of A, B and C being   respectively is given by

The positon vector of the centroid of a tetrahedron if the pv’s of its angular vertices are are given by  

Note that this is also the point of concurrency of the lines joining the vertices to the centroids of the opposite faces and is also called the centre of the tetrahedron. In case the tetrahedron is regular it is equidistant from the vertices and the four faces of the tetrahedron .

12. Vector Triple Product : Let be any three vectors, then the expression  is a vector & is called a vector triple product .

Geometrical interpretation of

Consider the expression  which itself is a vector, since it is a cross  product  of  two  vectors   Now  is  a vector perpendicular to the plane containing   but is a vector perpendicular to the plane   therefore  is  a vector lies in the plane of  and perpendicular to  Hence we can express   in terms of   i.e. where x & y are scalars .

13. Linear combinations / Linearly Independence and Dependence of Vectors :
Given a finite set of vectors  then the vector  

is called a linear combination of  for any x, y, z ...... ∈ R. We have the following results :

(a) Fundamental theorem in plane :  Let  be  non zero ,  non collinear vectors . Then any vector  coplanar with  can be expressed uniquely as a linear combination of   There exist some unique x,y ∈ R such that

(b) Fundamental theorem in space : Let  be non−zero, non−coplanar vectors in space. Then  any vector  can be uniquily expressed as a linear combination of

There exist some unique x,y ∈ R such that

(c) If  are  n  non zero vectors,  & k₁, k₂, .....k_n  are n  scalars & if the linear  combination

Linearly independent vectors

(d) If   are  not Linearly independent  then  they  are said  to be Linearly dependent vectors  &  if  there  exists at least  one kr ≠ 0 then 

are said to be linearly dependent .

Note:

If  then  is expressed as a linear combination of vectors  form a linearly dependent set of vectors. In general , every set of four vectors is a linearly dependent system.  are Linearly independent set of vectors. For

Two vectors  are linearly dependent ⇒  is parallel to linear dependence of  Conversely if  then  are linearly independent  .

If  three vectors  are linearly dependent, then they are coplanar i.e.  

conversely, if   then the vectors are linearly independent.

14. Coplanarity of  vectors:
Four points A, B, C, D with position vectors  respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that  where,  x + y + z + w = 0.

15. Reciprocal system of vectors:
If   are two sets of non coplanar vectors such that

then the two systems are called Reciprocal System of vectors. 

Note:

16. Equation of a plane 
(a) The  equation  represents  a  plane  containing  the  point  with

 where  is a vector normal to the plane . is the general equation of a plane.

(b) Angle between the 2 planes is the angle between 2 normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane.

17. Application of vectors:
(a) Work done against a constant force  over a displacement  is defined as

(b) The  tangential  velocity of  a  body moving  in a circle is given by   where  is the pv of the point P.

(c) The moment of  about ’O’ is defined as  is the pv  of  P  wrt  ’O’.  The direction  of  is  along  the  normal  to  the plane OPN   such   that   

 form  a right  handed system.

(d) Moment of the couple =  where  are  pv’s of the point of the application of the forces .