ADDITION OF VECTORS questions and answers

JEE MAINS - VOL - I ADDITION OF VECTORS


ADDITION OF VECTORS
 Free Vector: When a vector is specified by
SYNOPSIS not fixing initial point or terminal point or both, is
called a free vector, i.e., a free vector does not
 Definition of Vector and Scalar: have specific initial point or terminal point or both.
 Scalar: A physical quantity which has only
Note: (i) A vector a means we are free to
magnitude is called a scalar.
choose initial or terminal point anywhere. Once
Examples: Length,volume, speed, time. initial point is fixed at A then terminal point is
 Vector: A vector is a physical quantity which
has both magnitude and direction . uniquely fixed at B such that AB a
Geometrically a directed line segment is called a (ii) A free vector is subjected to parallel
vector. displacement without changing the magnitude and
Examples: Force, Velocity, acceleration. direction.
Note: All real numbers are scalars. (iii) In general vectors are considered to be free
 Notation: Vectors are denoted by directed line vectors unless they are localised.
 Let a be a nonzero vector then
segments such as AB CD.... or by a , b ... If
a
AB is a vector then A is called initial point and (i) Unit vector in the direction of a  a
B is called the terminal point or final point and
(ii) Unit vector in the direction opposite to that of
the direction of AB is from A to B. The magnitude
a
a is 
of AB is denoted by AB or AB and, is the a
distance between the points A and B. a
 Types of Vectors: (iii) Unit vectors parallel to a   a .
 Position Vector: Let O be a fixed point (called (iv) The vectors having magnitude m units and
the origin ) and let P be any point. If OP  r
ma
then r is called the position vector of P with parallel to a   .
a
respect to O.
 Null Vector: A vector having zero magnitude WE-1: a  i  2 j  2k and b  3i  6 j  2k ,
(arbitrary direction) is called the null (zero) vector.
then vector in the direction of a and having
It is denoted by 0 .
Note (i ) A zero vector can be regarded as magnitude as b is
having any direction for all mathematical Sol: The required vector is
calculations.
(ii) A non-zero vector is called a proper vector b 7
 Unit Vector: A vector whose magnitude is a
a
3
 i  2 j  2k 
equal to one unit is called a unit vector.
 Localised vector :A vector is a localised  Equal Vectors: Two vectors a and b are
vector, if the vector is specified by giving either equal if they have the same magnitude
initial point or terminal point (or) if a vector is
specified by fixing atleast one of its ends is called i.e a  b  and they are in the same
a localised vector.
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, ADDITION OF VECTORS JEE MAINS - VOL - I

direction. a  b represents the diagonal of the
 Let a and b be the position vectors of the parallelogram through the common points. It is
points A and B respectively. Then AB  known as parallelogram law of vector addition.
(position vector of B) - (position vector of A) i.e., OC  OA  OB  a  b
i.e., AB  b  a  Properties of Addition of Vectors:
 Co-Initial Vectors:Vectors having the same i) Addition of vectors is commutative
initial point are called coinitial vectors. The vectors i.e., a  b  b  a
ii) Addition of vectors is associative
OA, OB, OC... are coinitial vectors.
 Co-Terminal Vectors: The vectors having the i.e., a  b   c  a  b  c 
same terminal point are called the co-terminal iii) There exists a vector 0 such that
vectors. The vectors AO, BO, CO... are a  0  0  a  a . Then 0 is called the
co-terminal vectors. additive identity vector.
 Addition of Vectors: iv) To each vector a there exists a vector a
 Triangle law of Vector Addition: such that a  a   a   a  0 .Then a
A is called the additive inverse of a .
 Scalar Multiplication of Vectors: Let
a a b a be a nonzero vector and let m be a scalar. Then
m a is a scalar multiplication of a by m .
Note: i) The direction of m a is along a if m>o.
ii) The direction of m a is opposite to that of a if
B C m<o.
b
 Properties of Scalar Multiplication
If AB  a and BC  b are two non-zero of Vectors: If a , b are vectors & m, n
vectors are represented by two sides of a triangle are scalars, then the magnitude (length) of m a
ABC then the resultant (sum) vector is given by is m times that of a .
 
the closing side AC of the triangle in opposite Note: (i) m(na )  n(ma )  (mn) a
(ii) (m  n)a  ma  na
direction. i.e., AC  AB  BC  a  b
 Parallelogram Law of Vector (iii) m( a  b )  ma  mb
Addition:  If a and b are any two vectors,then

B (i) a  b  a  b
C

b (ii) a  b  a  b
a b
(iii) a  b  a  b
O a A (iv) If a and b are like vectors, then
If a and b are two adjacent sides of the a b  a  b
parallelogram, then their sum (resultant)

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,JEE MAINS - VOL - I ADDITION OF VECTORS

 Components of a space Vector: WE-2: If a , b are the position vectors of A,B
respectively and C is a point on AB produced
Z such that AC = 3AB, then the position vector
of C is
Sol: Let the position vector for C be c , then B divides
AC internally in the ratio 1:2, therefore
k
2a  c
P(x, y, z) b  c  3b  2a .
2 1
Y  The position vector of the centroid G of the
i O j
a b c
triangle ABC with vertices a , b , c is
3
X aa  bb  cc
and the incentre I  , where
abc
 Let i , j , k be unit vectors acting along the positive
a  BC , b  CA and c  AB .
directions of x, y and z axes respectively, then
 In ABC if D, E, F are the mid points of the
position vector of any point P in the space is
sides BC, CA, AB respectively and G is the
OP  xi  yj  zk . Here (x, y, z) are called
centroid then (i) GA  GB  GC  0
scalar components of vector OP along respective
(ii) AD  BE  CF  0 .
axes and xi , yj , zk are called vector
(iii) OA  OB  OC  OD  OE  OF  3OG
components of OP along respective axes &
 If a , b , c and d are the position vectors of the
OP  x 2  y 2  z 2 vertices A, B, C and D respectively of a
 Section Formula : If the position vectors of tetrahedron ABCD then the position vector of
the points A, B w.r.t. O are a and b and if the a b c d
its centroid is
point C divides the line segment AB in the ratio 4
m : n internally  m  0, n  0  , then the position  Angle between two vectors : If
OA  a , OB  b be two non-zero vectors
mb  na
vector of C is OC  . and AOB  , 00   1800 is defined as the
mn
 If C is an external point that divides A  a  , B  b  angle between a and b and is written as  a , b  .
in the ratio m : n externally then
B
mb  na
OC 
mn
 m  n  and  m, n  0  b
180O-
 The point C (mid point) divides A  a  , B  b  in 
A A
a b a  O a
the ratio 1:1 , then OC  .
2 180O-
 Points of trisection : Two points which divide b
a line segment in the ratio 1: 2 and 2 :1 are called
B
the points of trisection.
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, ADDITION OF VECTORS JEE MAINS - VOL - I

Note: If a  0 or b  0 , then angle between k  R  0
a and b is undefined.  Some important results: If l , m, n are the
Note: (i)  a , b   0  a and b are like direction cosines of a line, then
vectors.
l 2  m2  n2  1 .

(ii)  a , b    a and b are orthogonal  If OP  r and P is the ordered triad  x, y, z 
2
vectors. then x  r cos   lr , y  r cos   mr and
 
(iii) a , b    a and b are unlike vectors. z  r cos   nr .
 The direction cosines of the vectors i , j , k are
(iv)  a , b    b , a  and  a , b     a , b 
respectively 1, 0, 0  ,  0,1, 0  ,  0, 0,1 .
(v)  ma , nb    a , b  (if m,n have same signs).
 Linear Combination of Vectors:
(vi)  ma , nb   180   a , b 
0

 Linear Combination: Let a1 , a2 ,..., an be n
(if m,n haveopposite signs).
vectors and let r be any vector. Then
 Right handed and left handed triads
r  x1 a1  x2 a2  x3 a3  ....  xn an is called
Let OA  a , OB  b , OC  c be three non-
coplanar vectors. Viewing from the point C, if the a linear combination of the vectors a1 , a2 ,..., an .

rotation of OA to OB does not exceed angle  Collinear Vectors: Vectors which lie on a line
or on parallel lines are called collinear vectors
180 in anti-clock sense, then a , b , c are set to (whatever be their magnitudes).
form a right handed system of vectors and we
Note:(i) Two vectors a and b are collinear if
say simply that ( a , b , c ) is right handed vector
and only if a  mb or b  na where m ,n are
triad. If ( a , b , c ) is not a right handed system, scalars (real numbers).
then it is called left handed system.
(ii) Let the vectors a  a1 i  a2 j  a3 k ,
B A
b a b  b1 i  b2 j  b3 k are collinear if and only if
O
O
a A a1 a 2 a3
c c b B  
b1 b2 b3
C C
 Direction cosines and Direction (iii) The Vectors a , b are collinear vectors ifff
ratios of a vector: Let i , j , k be an unit a,b   0 0
or180
vector traid in the right handed system and r is a
 Let a and b be two non collinear vectors and
vector. If    r , i  ,    r , j  and
let r be any vector coplanar with them.Then
   r , k  , then cos  , cos  , cos  are called
r  xa  yb and the scalars x and y are unique
the direction cosines of r denoted by l , m, n
in the sense that if r  x1a  y1b and
respectively.The numbers proportional to
direction cosines of a given vector, i,e., kl , km, kn r  x2 a  y2b then x1  x2 and y1  y2 .
are called the direction ratios of that vector for
The vector equation r  xa  yb implies that
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