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Important Terms, Definitions & Formulae

or a . Its magnitude (or modulus) is AB

or a otherwise, simply AB or a .

Vectors are denoted by symbols such as a or a or a.

[Pictorial representation of vector]

02. Initial and Terminal points: The initial and terminal points mean that point from which the vector originates and terminates respectively.

03. Position Vector: The position vector of a point say P( , , )x y z is OP ˆˆ ˆ

r x y zi j k and the

magnitude is 2 2 2 r x y z . The vector OP ˆˆ ˆ

r x y zi j k is said to be in its component form.

Here , ,x y z are called the scalar components or rectangular components of r

and ˆˆ ˆ, ,x y zi j k are the

vector components of r

along x-, y-, z- axes respectively.

Also, AB Position Vector of B Position Vector of A

. For example, let

1 1 1A( , )x y ,z and 2 2 2B( , )x y ,z . Then, 2 2 2 1 1 1ˆ ˆˆ ˆ ˆ ˆAB ( ) ( )

x i y j z k x i y j z k .

Here ,ˆ ˆi j and k̂ are the unit vectors along the axes OX, OY and OZ respectively. (The

discussion about unit vectors is given later in the point 05(e).)

04. Direction ratios and direction cosines: If ˆˆ ˆ ,r x y zi j k

then coefficients of ˆˆ ˆ, ,i j k in r

i.e., , ,x y z

are called the direction ratios (abbreviated as d.r.’s) of vector r

. These are denoted by , ,a b c (i.e.

, , a x b y c z ; in a manner we can say that scalar components of vector rsame).

Also, the coefficients of ˆˆ ˆ, ,i j k in r̂ (which is the unit vector of r

) i.e., 2 2 2

x

x y z,

2 2 2

y

x y z,

2 2 2

z

x y z are called direction cosines (which is abbreviated as d.c.’s) of vector r

.

These direction cosines are denoted by l, m, n such that cos , cos , cos l m n and

2 2 2 2 2 21 cos cos cos 1 l m n .

It can be easily concluded that cos , cos , x y

l mr r

cos z

nr

.

Therefore, cos cosˆ ˆˆ ˆ ˆ ˆ(cos ) r mr nrlri j k r i j k . (Here

r r .)

[See the OAP in Fig.1] Angles , , are made by the vector r

with the positive directions of x, y, z-axes

respectively and these angles are known as the direction angles of vector r

).

For a better understanding, you can visualize the Fig.1.

05. TYPES OF VECTORS a) Zero or Null vector: Its that vector whose initial and terminal points are coincident. It is denoted

by 0

. Of course its magnitude is 0 (zero). Any non-zero vector is called a proper vector. b) Co-initial vectors: Those vectors (two or more) having the same initial point are called the co- initial vectors.

Vector Algebra Formulae For

BASIC ALGEBRA OF VECTORS

By Mir Mohammed Abbas II PCMB 'A'

01. Vector - Basic Introduction: A quantity having magnitude as well as the direction is called vector. It is

denoted as AB

and its d.r.’s both are the

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c) Co-terminous vectors: Those vectors (two or more) having the same terminal point are called the co-terminous vectors. d) Negative of a vector: The vector which has the same magnitude as the r

but opposite direction. It

is denoted by r

. Hence if, AB BA

r r . That is AB BA, PQ QP

etc.

e) Unit vector: It is a vector with the unit magnitude. The unit vector in the direction of vector r

is

given by ˆ rr

r such that 1r

. So, if ˆˆ ˆr x y zi j k

then its unit vector is:

2 2 2 2 2 2 2 2 2

ˆˆ ˆˆ x y zi j k

x y z x y z x y zr

.

Unit vector perpendicular to the plane of a and

b is:

a b

a b.

f) Reciprocal of a vector: It is a vector which has the same direction as the vector r

but magnitude

equal to the reciprocal of the magnitude of r

. It is denoted as 1r . Hence 1 1r

r

.

g) Equal vectors: Two vectors are said to be equal if they have the same magnitude as well as direction, regardless of the positions of their initial points.

Thus

a b

a b

and have same direction

a b .

Also, if 1 2 3 1 2 3 1 1 2 2 3 3ˆ ˆˆ ˆ ˆ ˆ , ,

a b a i a j a k b i b j b k a b a b a b .

h) Collinear or Parallel vector: Two vectors a and

b are collinear or parallel if there exists a non-

zero scalar such that a b

.

It is important to note that the respective coefficients of ˆˆ ˆ, ,i j k in a and

b are proportional

provide they are parallel or collinear to each other.

Consider 1 2 3 1 2 3ˆ ˆˆ ˆ ˆ ˆ,

a a i a j a k b b i b j b k , then by using a b

, we can conclude

that: 31 2

1 2 3

aa a

b b b .

The d.r.’s of parallel vectors are same (or are in proportion).

The vectors a and

b will have same or opposite direction as is positive or negative.

The vectors a and

b are collinear if 0

a b .

i) Free vectors: The vectors which can undergo parallel displacement without changing its magnitude and direction are called free vectors.

06. ADDITION OF VECTORS a) Triangular law: If two adjacent sides (say sides AB and BC) of a triangle ABC are represented by

a and

b taken in same order, then the third side of the triangle taken in the reverse order gives the

sum of vectors a and

b i.e., AC AB BC

AC

a b . See Fig.2.

Also since AC CA AB BC CA AA 0

.

And AB BC AC

AB BC AC 0 AB BC CA 0

.

b) Parallelogram law: If two vectors a and

b are represented in magnitude and the direction by the

two adjacent sides (say AB and AD) of a parallelogram ABCD, then their sum is given by that

diagonal of parallelogram which is co-initial with a and

b i.e., OC OA OB

. For the

illustration, see Fig.3.

Multiplication of a vector by a scalar Let a be any vector and k be any scalar. Then the product

ka is defined as a vector whose

magnitude is k times that of a and the direction is

(i) same as that of a if k is positive, and (ii) opposite to that of

a if k is negative.

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MATHEMATICS – List Of Formulae for Class XII By Mohammed Abbas (II PCMB 'A')

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07. PROPERTIES OF VECTOR ADDITION

• Commutative property: a b b a

.

Consider 1 2 3ˆˆ ˆa a i a j a k

and 1 2 3

ˆˆ ˆb b i b j b k

be any two given vectors.

Then 1 1 2 2 3 3ˆˆ ˆa b a b i a b j a b k b a

.

• Associative property: a b c a b c

• Additive identity property: 0 0a a a

.

• Additive inverse property: ( ) 0 ( )

a a a a .

08. Section formula: The position vector of a point say P dividing a line segment joining the points A and B

whose position vectors are a

and b

respectively, in the ratio m:n

(a) internally, is OP

mb na

m n

(b) externally, is OP

mb na

m n.

Also if point P is the mid-point of line segment AB then, OP2

a b.


01. PRODUCT OF TWO VECTORS

a) Scalar product or Dot product: The dot product of two vectors a

and b

is defined by,

. cosa b a b

where θ is the angle between a

and b

, 0 . See Fig.4.

Consider 1 2 3 1 2 3ˆ ˆˆ ˆ ˆ ˆ,a a i a j a k b b i b j b k

. Then 1 1 2 2 3 3.a b a b a b a b

.

Properties / Observations of Dot product

• ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ. cos0 1 . 1 . .i i i i i i j j k k .

• ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ. cos 0 . 0 . .2

i j i j i j j k k i

.

• . R,

a b where R is real number i.e., any scalar.

• . .a b b a

(Commutative property of dot product).

• . 0a b a b

.

• If θ 0= then, .

a b a b . Also 2 2.

a a a a ; as θ in this case is 0 .

Moreover if θ = then, .

a b a b .

• . . .a b c a b a c

(Distributive property of dot product).

• . . .a b a b a b

.

• Angle between two vectors a

and b

can be found by the expression given below:

.

cos

a b

a b

PRODUCT OF VECTORS

DOT PRODUCT & CROSS PRODUCT

List Of Formulae By Mohammed Abbas www.mohammedabbas27.wordpress.com


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or, 1 .cos

a b

a b

.

• Projection of a vector a

on the other vector say b

is given as .a b

b

i.e., ˆ.a b

.

This is also known as Scalar projection or Component of a

along b

.

• Projection vector of a

on the other vector say b

is given as . ˆ.

a bb

b

.

This is also known as the Vector projection.

• Work done W in moving an object from point A to the point B by applying a force F

is

given as W F.AB

.

b) Vector product or Cross product: The cross product of two vectors a

and b

is defined by,

ˆsina b a b

, where is the angle between the vectors a

and b

, 0 and ̂ is a

unit vector perpendicular to both a

and b

. For better illustration, see Fig.5.

Consider 1 2 3 1 2 3ˆ ˆˆ ˆ ˆ ˆ,a a i a j a k b b i b j b k

.

Then, 1 2 3 2 3 3 2 1 3 3 1 1 2 2 1

1 2 3

ˆˆ ˆ

ˆˆ ˆ

i j k

a b a a a a b a b i a b a b j a b a b k

b b b

.

Properties / Observations of Cross product

• ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆsin 0. 0 0i i i i j i i j j k k

.

• ˆ ˆˆ ˆ ˆ ˆ sin .2

i j i j k k

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ, ,i j k j k i k i j .

• Fig.6 at the end of chapter can be considered for memorizing the vector product of ˆˆ ˆ, ,i j k .

• a b

is a vector c

(say) and this vector c

is perpendicular to both the vectors a

and b

.

• ˆsin sin

a b a b a b i.e., sin

a b ab .

• 0a b

//a b

or, 0, 0

a b .

• 0a a

.

•

a b b a (Commutative property does not hold for cross product).

• ;a b c a b a c b c a b a c a

(Distributive property of the vector product or cross product).

• Angle between two vectors a

and b

in terms of Cross-product can be found by the

expression given here: sin

a b

a b

or, 1sin

a b

a b

.

• If a

and b

represent the adjacent sides of a triangle, then the area of triangle can be

obtained by evaluating 1

2

a b .



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• If a

and b

represent the adjacent sides of a parallelogram, then the area of parallelogram

can be obtained by evaluating

a b .

• If p

and q

represent the two diagonals of a parallelogram, then the area of

parallelogram can be obtained by evaluating 1

2 p q .

02. Relationship between Vector product and Scalar product [Lagrange’s Identity]

Consider two vectors a

and b

. We also know that ˆsina b a b

.

Now, ˆsin

a b a b

sin

a b a b

2 22 2sin

a b a b

2 22 2

2 2 22 2 2

1 cos

cos

a b a b

a b a b a b

22 22

2 2 22

cos

.

a b a b a b

a b a b a b

or, 2 22 2

.a b a b a b

.

Note that 2 . .

. .

a a a ba b

a b b b

. Here the RHS represents a determinant of order 2.

03. Cauchy- Schwartz inequality:

For any two vectors a

and b

, we always have .a b a b

.

Proof: The given inequality holds trivially when either 0

a or 0 b i.e., in such a case . 0a b a b

.

So, let us check it for 0

a b .

As we know, . cosa b a b

2 22 2. cosa b a b

Also we know 2cos 1 for all the values of .

2 22 22cosa b a b

2 22

.a b a b

.a b a b

. [H.P.]

04. Triangle inequality:

For any two vectors a

and b

, we always have

a b a b .

Proof: The given inequality holds trivially when either 0

a or 0 b i.e., in such a case we have

If a and

b represent the adjacent sides of a parallelogram, then the

diagonals 1

d and 2

d of the parallelogram are given as:

1 d a b , 2

d b a .



0

a b a b . So, let us check it for 0

a b .

Then consider 2 22

2 .

a b a b a b

2 22

2 cos

a b a b a b

For cos 1 , we have: 2 cos 2a b a b

2 22 2

2 cos 2a b a b a b a b

22

a b a b

a b a b . [H.P.]


01. SCALAR TRIPLE PRODUCT:

If a

,b

and c are any three vectors, then the scalar product of

a b with

c is called scalar triple

product of a

,b

and c .

Thus, ( ).

a b c is called the scalar triple product of a

,b

and c .

Notation for scalar triple product: The scalar triple product of a

,b

and c is denoted

by [ ]

a b c . That is, ( ). [ ]

a b c a b c .

Scalar triple product is also known as mixed product because in scalar triple product, both the signs of dot and cross are used.

Consider 1 2 3 1 2 3 1 2 3ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ, ,

a a i a j a k b b i b j b k c c i c j c k .

Then,

1 2 3

1 2 3

1 2 3

[ ]

a a a

a b c b b b

c c c

.

Properties / Observations of Scalar Triple Product

• ( ). .( )

a b c a b c . That is, the position of dot and cross can be interchanged without

change in the value of the scalar triple product (provided their cyclic order remains the same).

• [ ] [ ] [ ]

a b c b c a c a b . That is, the value of scalar triple product doesn’t change

when cyclic order of the vectors is maintained.

Also [ ] [ ]; [ ] [ ]

a b c b a c b c a b a c . That is, the value of scalar triple

product remains the same in magnitude but changes the sign when cyclic order of the vectors is altered.

• For any three vectors a

, b

, c and scalar , we have [ ] [ ]

a b c a b c .

• The value of scalar triple product is zero if any two of the three vectors are identical.

That is, [ ] 0 [ ] [ ]

a a c a b b a b a etc.

• Value of scalar triple product is zero if any two of the three vectors are parallel or collinear.

• Scalar triple product of ˆ ˆ,i j and k̂ is 1 (unity) i.e., ˆˆ ˆ[ ] 1i j k .

SCALAR TRIPLE PRODUCT OF VECTORS

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• If [ ] 0

a b c then, the non-parallel and non-zero vectors a

, b

and c are coplanar.

Volume Of Parallelopiped

If a

, b

and c represent the three co-terminus edges of a parallelopiped, then its

volume can be obtained by: [ ] ( ).

a b c a b c . That is,

( ). Base area of Parallelopiped Height of Parallelopiped on this basea b c

.

If for any three vectors a

, b

and c , we have [ ] 0

a b c , then volume of

parallelepiped with the co-terminus edges as a

, b

and c , is zero. This is possible

only if the vectors a

, b

and c are co-planar.



VARIOUS FIGURES RELATED TO THE VECTOR ALGEBRA

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