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 3 MATRICES

KEY CONCEPT INVOLVED

1. Matrices - A system of mn numbers (real or complex) arranged in a rectangular array of m rows and n
columns is called a matrix of order m × n. An m × n matrix (to be read as ‘m by n’ matrix)
An m × n matrix is written as
ª a11 a12 ........ a1n º
« »
«a 21 a 22 ........ a 2n »
A « # # ........ # »
« »
« # # ........ # »
¬«a m1 a m2 ........ a mn ¼»
The numbers a11, a12 etc are called the elements or entries of the matrix. If A is a matrix of order m × n, then
we shall write A = [aij]m × n where, aij represent the number in the i-th row and j-th column.
2. Row Matrix - A single row matrix is called a row matrix or a row vector. e.g. the matrix [a11, a12, ...... a1n] is
a row matrix.
ª a11 º
« »
« a 21 »
3. Column Matrix - A single column matrix is called a column matrix or a column vector. e.g. the matrix « # »
« »
is a m × 1 column matrix. ¬«a m1 ¼»

4. Order of a Matrix - A matrix having m rows and n columns is of the order m × n. i.e. consisting of m rows
and n columns is denoted by A = [aij]m × n.
5. Square Matrix - If m = n, i.e. if the number of rows and columns of a matrix are equal, say n, then it is called
a square matrix of order n.
6. Null or Zero Matrix - If all the elements of a matrix are equal to zero, then it is called a null matrix and is
denoted by Om × n or 0.
7. Diagonal Matrix - A square matrix, in which all its elements are zero except those in the leading diagonal is
called a diagonal matrix, thus in a diagonal matrix, aij = 0, if i z j, e.g. the diagonal matrices of order 2 and 3
ªK 0 0 º
ª K1 0 º « 1 »
are « » , « 0 K2 0 »
¬ 0 K 2¼ «
¬0 0 K 3 »¼
8. Scalar Matrix - A square matrix in which all the diagonal element are equal and all other elements equal to
zero is called a scalar matrix.
ªK 0 0 º
i.e. in a scalar matrix aij = k for i = j and aij = 0 for i z j. Thus « 0 K 0 » is a scalar matrix.
« »
«¬ 0 0 K »¼

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 9. Unit Matrix or Identity Matrix - A square matrix in which all its diagonal elements are equal to 1 and all
other elements equal to zero is called a unit matrix or identity matrix.
1 0 0
1 0
e.g. a unit or identity matrix of order 2 and 3 are and 0 1 0 respectively..
0 1
0 0 1
10. Upper triangular Matrix - A square matrix A whose elements aij = 0 for i > j is called an upper triangular
matrix.
11. Lower triangular Matrix - A square matrix A whose elements aij = 0 for i < j is called a lower triangular
matrix.
12. Equal Matrices - Two matrices A and B are said to be equal, written as A = B if
(i) they are of the same order i.e. have the same number of rows and columns, and
(ii) the elements in the corresponding places of the two matrices are the same.
13. Transpose of a matrix - Let A be a m × n matrix then the matrix of order n × m obtained by changing its rows
into columns and columns into rows is called the transpose of A and is denoted by A or AT.
14. Negative of Matrix - Let A = [aij]m × n be a matrix. Then the negative of the matrix A is defined as the matrix
[–aij]m ×n and is denoted by –A.
15. Symmetric Matrix - a square matrix A is said to be symmetric if A = A
Thus a square matrix A = [aij] is symmetric if A = [aij] is symmetric if aij = – aji for all values of i and j.
16. Skew-Symmetric Matrix - A square matrix A is said to be skew-symmetric if A = – A Thus a square matrix
A = [aij] is skew-symmetric if aij = – aji for all values of i and j.
In particular aii = – aii 2aii = 0 aii = 0 i.e. all diagonal elements of a skew-symmetric matrix are o.
17. For any square matrix A with real number entries, A + A is a symetric matrix and A – A is a skew symetric
matrix.
18. Any square matrix can be expressed as the sum of a symetric and a skew symetric matrix.
1 1 1
If A be a square matrix, then we can write A (A A ) (A A ) , here (A A ) is symetric matrix
2 2 2
1
and (A A ) is skew symetric matrix.
2
19. Addition of Matrices - Let there be two matrices A and B of the same order m × n. then the sum denoted
by A + B is defined to be the matrix of order m × n obtained by adding the corresponding elements of
A and B.
Thus if A = [aij]m × n and B = [bij]m × n then A + B = [aij + bij]m × n
20. Scalar Multiplication of a Matrix - Let A = [aij]m × n be a matrix and K is a scalar. Then the matrix obtained
by multiplying each element of matrix A by K is called the scalar multiplication of matrix A by K and is
denoted by KA or AK.
21. Multiplication of Matrices - Product of two matrices exists only if number of column of first matrix is equal
to the number of rows of the second. Let A be m × n and B be n × p matrices. Then the product of matrices
A and B denated by A.B is the matrix of order m × p whose (i, j)th element is obtained by adding the
products of corresponding elements of ith row of A and jth column of B.
22. Elementary Row Operations - The operations known as elementary row operations on a matrix are-
(i) The interchange of any two rows of a matrix. (The notations Ri Rj is used for the interchange of
the i-th and j-th rows.)
(ii) The multiplication of every element of a row by a non-zero element (constant).
(The notations K.Ri is used for the multiplication of every element of i-th row by a constant K.
(iii) The addition of the elements of a row, the product of the corresponding elements of any other row by
any non-zero constant. (The notation Ri + K.Rj is generally used for addition to the elements of
i-th row to the element of j-th row multiplied by the constant K (K 0))
23. Invertible matrices - If A is a square matrix of order m, and if there exists another square matrix B of the
same order m, such that AB = BA = I, then B is called the Inverse matrix of A and it is denoted by A–1. In
that care A is said to be invertible.

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24. If A and B are invertible matrices of the same order, then (AB)–1 = B–1. A–1.
25. Inverse of a matrix by elementry operations - Let X, A and B be matrices of, the same order such that X =
AB. In order to apply a sequence of elementry row operations on the matrix equation X = AB, we will apply
these row operations simultaneously on X and on the first matrix A of the product AB on RHS.
Similarly, in order to apply a sequence of elementry column operations on the matrix equation X = AB, we
will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS.
In view of the above discussion, we conclude that if A is a matrix such that A–1 exists, then to find A–1
using elementry row operations, write A = IA and apply a sequence of row operation on A = IA till we get,
I =BA. The matrix B will be the inverse of A. Similarly, if we with to find A–1 using column operations, then,
write A = AI on A = IA till we get, I = BA. The matrix and apply a sequence of column operations on A = AI
till we get, I = AB.
Remark - In case, after applying one or more elementry row (column) operations on A = IA (A = AI). If we
obtain all zero in one or more rows of the matrix A on L.H.S., that A–1 does not exist.

CONNECTING CONCEPTS
1. The elements aij of a matrix for which i = j are called the diagonal elements of a matrix and the line along
which all these elements lie is called the principal diagonal or the diagonal of the matrix.
2. Properties of transpose of the matrices-
(i) (A + B)c = Ac + Bc
(ii) (KA)c = KAc, where K is constant
(iii) (AB)c = BcAc
(iv) Ac c $
3. Properties of Matrix addition-
(i) Matrix Addition is Commutative - If A and B be two m × n matrices, then A + B = B + A
(ii) Matrix Addition is Associative - If A, B and C be three m × n matrices, then
(A + B) + C = A + (B + C)
4. Properties of Multiplication of a Matrix by a Scalar-
(i) If K1 and K2 are scalars and A be a matrix, then (K1 + K2) A = K1 A + K2 A.
(ii) If K1 and K2 are scalars and A be a matrix, then K1 (K2 A) = (K1 K2) A.
(iii) If A and B are two matrices of the same order and K, a scalar, then K (A + B) = KA + KB.
(iv) If K1 and K2 are two scalars and A is any matrix then (K1 + K2) A = K1 A + K2 A.
(v) If A is any matrix and K be a scalar.
then (–K) A = – (KA) = K (–A).
5. Properties of Matrix Multiplication -
(i) Associative law for Multiplication - If A, B and C be three matrices of order m × n and n × p and
p × q, respectively, then (AB) C = A (BC).
(ii) Distributive Law - If A, B, C be three matrices of order m × n, n × p and n × q respectively.
then A ˜ (B + C) = A ˜ B + A ˜ C
(iii) Matrix Multiplication is not commutative.
i.e. A˜ Bz B˜A
(iv) The existence of multiplicative Identity : For every square matrix A, there exists an identity matrix of
same order such that IA = AI = A.
6. If A be any n × n square matrix, then
A ˜ (Adj A) = (Adj A) ˜ A = |A|. In
where In is an n × n unit matrix
7. (i) Only square matrix can have inverse
(ii) The matrix B = A–1, will also be a square matrix of same order A.
(iii) The square matrix A is said to be invertible if A–1 exists.
8. Every invertible matrix possesses a unique inverse.

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Permutations and Combinations

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Permutations and Combinations

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 n
P= ¦ x p = E (X)
i 1
i i

The mean of a random variables X is also called the expected value of X denoted by E (x).
8. Variance of a Random Variable – let X be a random variable with possible values x1 x2, ˜˜˜˜˜˜˜ xn occur with
probabilities are p1, p2 ˜˜˜˜˜˜˜pn respectively.
let P = E (X) be the mean of X. The variance of X denoted by var (X) or V 2x is defined as
n

Var (X) or V 2x = ¦
i 1
(xi - P)2 pi = E (xi – P)2 = E (X2) – [E (X)]2

Standard Deviation, Vx = Var (X)

9. Probability function – The probability of x success is denoted by p (X = x) or P(x) and is given by P (x) =
nC qn – x px , x = 0, 1, 2, ˜˜˜˜˜˜˜n and q = 1 – P
x
The function P (x) is known as probability function of binomial distribution.

CONNECTING CONCEPTS
1. Partition of a sample space – A set of events E1, E2 , ˜˜˜˜˜˜˜ En is said to represent a partition of sample S
if
(i) Ei ˆ Fj = I if i z j , i, j = 1, 2,˜˜˜˜˜˜˜n
(ii) E1 ‰ E2 ‰ E3 ‰ ˜˜˜˜˜˜˜‰ En = S
(iii) P (Ei) > 0  i = 1,2, ˜˜˜˜˜˜˜n.
2. Theorem of total Probability – let D E1, E2, ˜˜˜˜˜˜˜En J be a partition of sample spaces and each event has
a non – zero probability If A be any event associated with S, then
P (A) = P (E1) P (A/E1) + P (E2) P (A/E2) + P (E3) P (A/E3) + ˜˜˜˜˜˜˜ + P (En) P (A/En)
n
P (A) = ¦ P (E ) P (A/E )
i 1
i i

3. A Few Terminologies –
(i) Hypothesis – When Baye’s theorem is applied the events E1, E2, ˜˜˜˜˜˜˜˜˜˜˜˜˜˜En are said to be hypothesis x.
(ii) Priori Porbability – The Porbabilites P (E1), P (E2) ˜˜˜˜˜˜˜˜˜˜˜˜˜˜P (En) are called priori.
(iii) Posteriori Porbabililty – The conditional probability P (Ei/A) is known as the posteriori probability
of hypothesis Ei where i = 1, 2, 3, ......, n
4. Probability Distribution of a Random Variable – let real numbers x1, x2, ˜˜˜˜˜˜˜˜˜˜˜˜˜˜xn be the possible value
of random variable and p1, p2,˜˜˜˜˜˜˜˜˜˜˜˜˜˜pn be probability corresponding to each value of the random
variable X. Then the probability distribution is
X: x1 x2 ˜˜˜˜˜˜˜˜˜˜˜˜˜˜ xn
P(X) : p1 p 2 ˜˜˜˜˜˜˜˜˜˜˜˜˜˜ pn.
(i) pi > 0 (ii) sum of porbabilites p1 + p2 + ˜˜˜˜˜˜˜˜˜˜˜˜˜˜ + pn = 1.
5. Binomial Distribution – Probability distribution of a number of successes, in an experiment consisting of
n Bernoulli trials are obtanied by Binomial expansioin of (q + p)n. Such a probability distribution is
X: 0 1 2 ˜˜˜˜˜˜˜˜˜˜˜˜˜˜ r ˜˜˜˜˜˜˜˜˜˜˜˜˜˜ n
P(X) : nC0 qn nC1 q n – 1 P n C2 q n – 2 P2 nCr q P n – r r nC Pn
n
This probability distribution is called binomial distribution with parameter n and p.
Where, p is the probability of success in each trial and q is the probability of not sucess in each trial.
? p+q=1,q=1–p

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 Probability
Revision Notes
Class - 11 Maths
Chapter 16 - Probability

Ɣ Probability is a numerical measure of the uncertainty of diverse phenomena.

It can range from 0 to 1 a positive value.
Ɣ The phrases 'probably', 'doubt’, ‘most probably', 'chances' and so on, all have
an element of ambiguity in them.
no. of favourable outcome
Ɣ Probability=
total no. of outcomes
Ɣ Approaches to Probability:
i. Statistical approach: Observation & data collection
ii. Classical approach: Only Equal probable events
iii. Axiomatic method: For real-life situations. It has a strong connection to set
theory.

Ɣ Random Experiments:
(i) There are multiple possible outcomes.
(ii) It is impossible to know the outcome ahead of time.

Ɣ Outcomes: An outcome is a probable result of a random experiment.

Ɣ Sample space refers to the set of all possible results of a random experiment.
The letter S stands for it. For example, in a coin toss, the sample space is Head,
Tail.
Each element of the sample space is referred to as a sample point. For
example, in a coin flip, the head is a sample point.

Class XI Maths 1
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Ɣ Event:
An event is a collection of favourable outcomes.
An event is defined as a subset E of a sample space S . For example, suppose you
get an unusual result when you roll a dice.

Ɣ Occurrence of an event:
The occurrence of an event E in a sample space S is said to have occurred if the
experiment's outcome is such that E . We say that the event E did not
happen if the outcome is such that E.

Ɣ Types of Event
i. Impossible and Sure Events
ii. Simple Event
iii. Compound Event

Ɣ Impossible and Sure Events:

Events that are both impossible and certain are described by the empty set
and the sample space S . The impossible event is denoted by , and the entire
sample space is referred to as the Sure Event.
For example, while rolling a dice, an impossible event is when the number is
greater than 6 and a sure event is when the number is less than or equal to 6.

Ɣ Simple (or elementary) event: A simple event has only one sample point of a
sample space.
There are exactly n simple occurrences in a sample space with n different
items. For example, if you roll a dice, a simple event could be receiving a four.

Class XI Maths 2
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Ɣ Compound Event: A compound event is one in which there are multiple
sample points.
For example, in the case of rolling a die, a simple event could be the event of
receiving a four.

Ɣ Algebra of Events:
i. Complementary Event
ii. Event ‘A or B’
iii. Event ‘A and B’
iv. Event ‘A but not B

Ɣ Complementary Event
Complementary event to A='not A'
Example: If event A= Event of getting odd number in throw of a die, that is
1, 3, 5 Then, Complementary event to A= Event of getting even number in
throw of a die, that is 2, 4, 6

A'= Ȧ:Ȧ S and A S A (Where S is the Sample Space)

Ɣ Event (A or B):
A B is known as the union of two sets A and B , it contains all those elements
which are present in either of the two sets.

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If the sets A and B correspond to two events in a sample space, then ‘A B’ is
the event ‘either A or B or both’. This event ‘A B’ is also called ‘A or B’
Event

Ɣ Event 'A and B' :

A B is known as the intersection of two sets A and B , it contains all those
elements which are common in both the two sets. i.e., which belong to both
‘A and B’ . If A and B are two events, then the set A A B denotes the event
‘A and B’ .
Thus, A B = {Ȧ : Ȧ A and Ȧ B}

Ɣ Event 'A but not B'

A–B is the set of all those elements which are in A but not in B . Therefore, the
set A–B may denote the event ‘ A but not B ’.

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A–B=A B'

Ɣ Mutually exclusive events

Events A and B are said to be mutually exclusive if the occurrence of one of
them precludes the occurrence of the other, i.e., if they can't happen at the same
time.
A die is thrown, for example. All even outcomes is event A , and all odd outcomes
is event B . Then A and B are mutually exclusive events, and they cannot happen
at the same time.
A sample space's simple events are always mutually exclusive.

Ɣ Exhaustive events:
Sample space contains lot of events together.
Example: A die is thrown.
Event A= All even outcome and event B= All odd outcome. Even A & B
together forms exhaustive events as it forms Sample Space.

Ɣ Axiomatic Approach to Probability:

Another way of explaining probability by using axioms are rules is called the
Axiomatic approach.
Let S be sample space of a random experiment. The probability P is a real

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Sequences and Series

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Sets, Relations & Functions

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Simple Inequalities

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 C

o
b
o
a+ o
b

A o B
a
This is known
JJJG as JJJG
the triangle law of vector addition.
Further AC = – CA
JJJG JJJG JJJG JJJG JJJG JJJG
AB  BC  CA ? AB  BC  CA = 0
when sidesJJJofG a triangle
JJJG JJJGABC are taken in order i.e. initial and terminal points coincides. Then
AB  BC  CA = 0
G G
(ii) Parallelogram law of vector addition – If the two vectors a and b are represented by the two adjacent
G G
sides OA and OB of a parallelogram OACB, then their sum a + b is represented in magnitude
JJJG JJJG JJJG
and direction by the diagonal OC of parallelogram through their common point O i.e., OA  OB OC
B C

o
o
b o+b
a

O o A
a
G
5. Multiplication of Vector by a Scalar – Let a be the given vector and O be a scalar, then product of O and
G G
a Oa
G G
(i) when O is +ve, then a and O a are in the same direction.
G G G G
(ii) whenO is –ve. then a and O a are in the opposite direction. Also O a O a.
6. Components of Vector – Let us take the points A (1, 0, 0), B (0, 1, 0) and C (0, 0, 1) on the coordinate axes
JJJG JJJG JJJG JJJG JJJG JJJG
OX, OY and OZ respectively. Now, | OA | = 1, | OB | = 1 and | OC | = 1, Vectors OA , OB and OC each

having magnitude 1 is known as unit vector. These are denoted by ˆi, ˆj and k̂ .
Z

k C (0, 0, 1)

j
0 Y
B
i (0, 1, 0)
A (1, 0, 0)

JJJG X
Consider the vector OP , where P is the point (x, y, z). Now OQ, OR, OS are the projections of OP on
coordinates axes.
JJJG JJJG JJJG
? O Q = x, O R = y, O S = z ? OQ xi, ˆ OR yjˆ , OS zkˆ

88
 Z

S
Zk
P (x, y, z)
o
r
yj
R Y

Q xi

X
JJJG JJJG G
Ÿ OP ˆ  yj,
xi, ˆ  zkˆ , | OP | x 2  y2  z2 |r|
JJJG
x, y, z are called the scalar components and x ˆi , yjˆ , zkˆ are called the vector components of vector OP .
7. Vector joining two points – Let P1(x1, y1, z1) and P2(x2, y2 z2) be the two points. Then vector joining the
JJJJG JJJG JJJG JJJJG
points P1 and P2 is P1P2 . Join P1, P2 with O. Now OP 2 OP1  P1P2 (by triangle law)
Z P2 (x2, y2, z2)
P1(x1, y1, z1)

O Y

X
JJJJG JJJG JJJG
? P1P2 OP 2  OP1
= (x 2 ˆi  y 2 ˆj  z 2 k)
ˆ  (x ˆi  y ˆj  z k)
1 1 1
ˆ (x 2  x1 ) ˆi  (y 2  y1 ) ˆj  (z 2  z1 ) kˆ
JJJJG
P1P2 (x 2  x1 )2  (y2  y1 )2  (z 2  z1 )2
8. Section Formula
PR m
(i) A line segment PQ is divided by a point R in the ratio m : n internally i.e.,
RQ n

m : n
o o
P(a) R (o
r) Q(b)
G G G
If a and b are the position vectors of P and Q then the position vector r of R is given by
G G
G mb  na
r
mn
G G
G ab
If R be the mid-point of PQ, then r
2
G G
(ii) when R divides PQ externally, i.e., | a & ˜b | nˆ

o
P (o
a) Q (b) R (o
r)

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 G G
G mb  na
Then r
mn
JJJG
9. Projection of vector along a directed line – Let the vector AB makes an angle T with directed line A .
JJJG JJJG G
Projection of AB on A = AB cos T AC p.
B

T
A o C
P
G G
The vector p is called the projection vector. Its magnitudes is b , which is known as projection of vector
JJJG JJJG JJJG
AB . The angle T between AB and AC is given by
JJJG JJJG JJJG JJJG
AB ˜ AC JJJG AB ˜ AC
cos T JJJG JJJG , Now projection AC = | AB | cos T JJJG
| AB || AC | | AC |
JJJG
JJJG § AC · JJJG G § pG · G
AB ˜ ¨¨ JJJG ¸¸ , If JJJ G G
AB a, then AC a ˜ ¨ G ¸ a ˜ pˆ
© | AC | ¹ ©| p|¹
G
G G G § b · G
Thus, the projection of a on b = a ˜ ¨¨ G ¸¸ a ˜ bˆ
©|b|¹
G
10. Scalar Product of Two Vectors (Dot Product) – Scalar Product of two vectors aG and b is defined as
G G G G
a ˜ b | a | | b | cos T
G
Where T is the angle between aG and b (0 dTdS)
G G G G G G G G
(i) when T = 0, then a ˜ b a b = ab Also a ˜ a a a a.a a 2

? iˆ ˜ iˆ ˆj ˜ ˆj
kˆ ˜ kˆ 1
S G G G G S
(ii) when T , then a ˜ b | a | | b | cos 0
2 2
ˆi ˜ ˆj ˆj ˜ kˆ kˆ ˜ ˆi 0
G
11. Vector Product of two Vectors (Cross Product) – The vector product of two non-zero vectors aG and b ,
G G
denoted by a u b is defined as
G G G G G G
a u b = | a | | b | sin T˜ nˆ , where T is the angle between a and b , 0 d T d S .
G G G G
Unit vector n̂ is perpendicular to both vectors a and b such that a ˜ b and n̂ form a right handed
orthogonal system. G G G G
(i) If T = 0, then a u b = 0, ?a u a 0
and ? ˆi u ˆi ˆj u ˆj kˆ u kˆ 0
G G G G
(ii) If T = – / 2 , then a u b = | a & b | nˆ
ˆi u ˆj k,
ˆ ˆj u kˆ ˆi, kˆ u ˆi ˆj
Also, ˆj u ˆi k,ˆ kˆ u ˆj ˆi and iˆ u kˆ ˆj

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 CONNECTING CONCEPTS

1. Direction
JJJG Cosines – Let OX, OY, OZ be the positive coordinate axes, P (x, y, z) by any point in the space.
Let OP makes angles D, E, J with coordinate, axes OX, OY, OZ. The angle D, E, J are known as direction
angles, cosine of these angles i.e.,
Z

C
z z)
J P (x, y,
0 E Y
x D y B
A

X
cos D, cos E, cos J are called direction cosines of line OP. these direction cosines are denoted by A , m, n
i.e., A = cos D, m = cos E, n = cos J
2. Relation Between, l, m, n and Direction Ratios –
JJJG
The perpendiculars PA, PB, PC are drawn on coordinate axes OX, OY, OZ reprectively. Let | OP | = r
x y
In ' OAP, ‘ A = 90°, cos D = A , ? x = A r , In ' OBP.. ‘ B = 90°, cos E = m ? y = mr
r r
z
In ' OCP, ‘ C = 90°, cos J = n , ? z = nr
r
Thus the coordinates of P may b expressed as ( A r, mr, nr)
Also, OP2 = x2 + y2 + z2, r2 = (lr)2 + (mr)2 + (nr)2 Ÿ A 2 + m2 + n2 = 1
Set of any there numbers, which are proportional to direction cosines are called direction ratio of the
vactor. Direction ratio are denoted by a, b and c.
The numbers A r mr and nr, proportional to the direction cosines, hence, they are also direction ratios of
JJJG
vector OP .
3. Properties of Vector Addition –
G G G G G G
1. For two vectors a, b the sum is commutative i.e., a  b b  a
G G G
2. For three vectors a, b and c , the sum of vectors is associative i.e.,
G G G G G G
(a + b) + c = a + (b + c)
G G G G G G G G
4. Additive Inverse of Vector a – If there exists vector – a such that a + (– a) = a – a = 0 then – a is called
G
the additure inverse of a
G G
5. Some Properties – Let a a1 iˆ  a 2 ˆj  a 3 kˆ and b b1 ˆi  b 2 ˆj  b 3 kˆ
G G
(i) a  b (a1 iˆ  a 2 ˆj  a 3 k)
ˆ  ( b ˆi  b ˆj  b k)
1 2 3
ˆ = (a1 + b1) î + (a2 + b2) ĵ + (a3 + b3) k̂
G G
(ii) a b or (a1 ˆi  a 2 ˆj  a 3 k)
ˆ ( b1 iˆ  b 2 ˆj  b3 k)
ˆ Ÿ a1 = b1, a2 = b2, a3 = b3
G
(iii) Oa O (a1 ˆi  a 2 ˆj  a 3 k) ˆ = (Oa ) iˆ  (Oa ) ˆj  (Oa ) kˆ
1 2 3
G G G G
(iv) a and b are parallel, if and only if there exists a non zero scalar O such that b Oa

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 i.e., b1 ˆi + b 2 ˆj + b3 kˆ = O (a1 ˆi + a 2 ˆj + a 3 k)
ˆ = (Oa ) iˆ  (Oa ) ˆj  (Oa ) kˆ
1 2 3

b1 b2 b3
? b1 = Oa1, , b2 = Oa2, b3 = Oa3 ? O
a1 a2 a3

6. Properties of scalar product of two vectors (Dot Product)

G G
a˜b
(i) cos T G G
|a ||b|
G G
If a a1 iˆ  a 2 ˆj  a 3 kˆ and b b1 ˆi  b2 ˆj  b3 kˆ
G G G
Then, a ˜ b (a1 ˆi  a 2 ˆj  a 3 k) ˆ ˜ (b ˆi  b ˆj  b k) ˆ , aG ˜ b = a b + a b + a b
1 2 3 1 1 2 2 3 3
G G
G a ˜b a1b1  a 2 b 2  a 3 b 3
G T G G
|a| a12  a 22  a 32 , | b | b12  b 22  b 32 ? cos | a || b | a12  a 22  a 32 ˜ b12  b 22  b32
G G G G G
(ii) aG ˜ b is commutative i.e., a ˜ b b ˜ a
G G G G G G
(iii) If D is a scalar, then (D a) ˜ b D (a ˜ b) a ˜ (D b)

7. Properties of Vector Product of two Vectors (Cross Product) –

G G G G
(i) (a) If a = 0 or b = 0, then a × b = 0
G G G G
(b) If a & b, then a u b = 0
G G
(ii) a u b is not commutative
G G G G G G G G
i.e. a u b b u a , but a u b b u a
G G G G
(iii) If a and b represent adjacent sides of a parallelogram, then its area | a u b |
G G 1 G G
(iv) If a, b represent the adjacent sides of a triangle, then its area = | a u b |
2
G G G G G G G
(v) Distributive property a u (b  c) a u b  a u c
G G G G G G
(a) If D be a scalar, then D (a u b) (D a) u b a u (D b)
G G
(b) If a a1ˆi  a 2 ˆj  a 3 k,
ˆ and b b ˆi  b ˆj  b kˆ
1 2 3

ˆi ˆj kˆ
G G
Then, a u b a1 a 2 a 3
b1 b 2 b3

G G
8 . If D1 E1 J are the direction angles of the vector a a1iˆ  a 2 ˆj  a 3 kˆ . Then direction cosines of a are
given as
a1 a2 a3
cos D = G , cos E = G , cos J = G
a a a

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 G G
9. Scalar Product of Two Vectors (Dot Product) – Scalar Product of two vectors a and b is defined as
G G G G
a ˜ b a b cos T
G G § S·
where T is the angle between a and b ¨ 0 d T  ¸
© 2¹
G G G G G G
(i) When T = 0, then a ˜ b a b . Also a ˜ a a˜a = a2
? ˆi ˜ ˆi ˆj ˜ ˆj
kˆ ˜ kˆ 1
S G G G G S
(ii) When T = , a ˜ b a b cos 0
2 2

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