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ADDITION OF VECTORS questions and answers
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JEE MAINS - VOL - I ADDITION OF VECTORSADDITION 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.
1
, 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
abc
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
mn
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
mn
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.
3
, 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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