( PRE FINAL DRAFT / TO CONTINUE )

**C O N T E N T S >**

**Definition****Unit vector****Basic set****Vector addition > Triangular / Parallelogram / Polygonal / Analytical****Vector subtraction****Lami’s theorem****Directional Cosines****Vector Multiplication > Vector / Cross product … Determinant & Component Methods****Scalar / Dot product****Vector components****Method of components****Vector classification****Scalar triple / mixed triple / boxed product.**

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**Physical quantities are divided into vectors and scalars**. A scalar quantity is denoted by a magnitude or real number-Eg.Temperature of a room,volume of a jug,density of steel or presssure of air in a tyre.In the case of vector quantities,a specific direction related to some underlying reference frame is needed to define it,in addition to a magnitude…Eg.displacement of a car and velocity of a body.The length of a vector is called it’s magnitude.It is indicated by using vertical bars or by using italics – | A| or *A .*If the length of a vector is one unit,then it’s a unit vector.

A component of a vector is the projection of the vector on an axis.The find this,we draw perpendicular lines from from the two ends of the vector to the axis.The process of finding the components is called resolving the vector.The component of a vector has the same direction [ along an axis ] as the vector.

A **unit vector** lacks both dimension and unit,it only specifies a direction.Regarding the relations among vectors,we have great freedom in choosing a co-ordinate system,as the relations does not depend on the location of the origin of the co-ordinate system or on the orientation of the axis.

**Signage of unit vectors** : Multiplication of two unit vectors in anti-clockwise direction gives the third vector + ve. Whereas, multiplication of any two unit vectors in clockwise direction gives the third unit vector -ve sign.

**Basic Set** **:** Consider two non zero vectors, a and b,where the direction of b is neither the same or the opposite to that of a .Let OA and OB be representations of a and b and P is the plane of the triangle, OAB.Now,any vector *v* whose representation OV lies in the plane P can be written as *v* = λ a + μ b *.* Here the co-efficients, λ and μ are unique.As the vectors have their directions parellel to the same plane,they are coplanar.Any vector coplanar with a and b can be expanded uniquely in the above form.Also, the expansion set cannot be reduced in number[ say,to a single vector ].Hence the pair of vectors ( a,b) is said to be a basic set for vectors lying in the plane P.If we are dealing with three co-planar vectrors,a,b and c , then it’s *v* = λ a + μ b + ν c , again a,b,c is a basic set..Eventhough any set of three non-coplanar vectors form a basic,the basic vectors are considered as orthogonal unit vectors. Here, the basic set is denoted by ( i,j,k ).

A vector does not necessarily have location,eventhough a vector may refer to a quantity defined at a particular point.Two vectors can be compared ,eventhough they measure a physical quantity defined at different points of space & time.Note- the applicability of vectors is largely based on **euclidean geometry…. that space is flat** [ for huge distances ].In such a scanario, we can compare two vectors at differtent points.

**A Vector must a) satisfy the parallelogram law of addition b) have a magnitude and direction independent of choice of co-ordinate system**.

* All quantities having magnitude and direction need not be vectors -* For eg, in the case of the rotation of a rigid body about a particular axis fixed in space, eventhough it has both magnitude ( angle of rotation ) and direction ( the direction of the axis ), but two such rotations do not combine according to the vector law of addition, unless the angles of rotation are infinitesimally small. Eg – when the two axes are perpendicular to each other and the rotations are by π/2 rad or 90º. Therefore,the commutative law of addition is not satisfied.

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**Vector addition** : Usually denoted by a **‘ tip to tail ‘** method,where a second vector is joined at it’s tail to the head or tip of the first vector. Now, a third vector is drawn from the tail of the first vector to the head of the second one.This third vector is the resultant vector.In case one needs to add a vector acting in the opposite direction,you have to just add the second vector with a -ve sign i.e, | r| = |a| + | -b|.

*Triangle law of Vector Addition : When two vectors are represented both in magnitude and direction by the two sides of a triangle,taken in the same order, their resultant is represented by the third side of the triangle ( both in magnitude & direction ) taken in the reverse direction.**Parallelogram law of Vector Addition : If two vectors can be represented both in magnitude and direction by two adjacent sides of a parallelogram, then the resultant is represented both in direction and magnitude by the corresponding diagonal of the parallelogram.*

here, A + B = B + A ,hence it is **commutative**, A+ (B+C) = (A+B)+C,hence **associative**. If k is a scalar,then k(A+B) = kA + kB, hence** distributive.**

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[ ** Note : In vector addition, |C| = | A | + | B| isn’t always true. **The magnitude of | A| + | B| depends on the magnitudes of | A| and | B| and on the angle between them.Only in the case where, **they are parellel** , is the magnitude of |C| = | A | + | B| . If they are anti-parallel,|C| = | A | – | B|. [ *As vectors are not ordinary numbers,ordinary multiplication is not applicable to them. ]*

- vector addition

*Polygonal law of Vector Addition* **:** *Used for adding more than two vectors. The vectors to be added are placed / drawn in such a way that the head of one is joined to the tail of the second and so on to the last vector.Finally a vector is drawn to connect the tail of the first vector to the head of the last vector. This law is applicable only if the vectors are acting on a particle at the same time.The added vectors represent an open polygon,whose resultant closes the polygon.*

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*Analytical method of Vector Addition* : Here, we are once again checking both the triangle and parallelogram law of vector addition using an analytical method.

- Triangle law – Consider two vectors | P| and |Q| acting simultaneously on a particle and inclined at an angle θ.The two vectors are represented both in magnitude and direction by the two sides | OA | and | OC | of Δ OAC taken in the same order, where | OC | is the resultant vector. Draw CN to connect with ON.For the right angled Δ CNO

OC² = R² = ON² + NC² = ( OA = AN )² + NC²

=> R² = ( P + AN )² + NC² …… ( 1 )

In the right angled triangle, Δ ANC ,

sin Θ = NC / Q => NC = Q sin Θ and cos Θ = AN / Q

=> AN = Q cos Θ

From eq.(1) R² = ( P + Q cos Θ )² + Q² sin² Θ

=> R² = P² + Q² cos² Θ + 2 PQ cos Θ + Q² sin² Θ

=> R² = P² + Q²( cos² Θ + sin² Θ ) + 2 PQ cos Θ

=> R² = P² + Q² + 2 PQ cos Θ ( as cos² Θ + sin² Θ = 1 )

=> R = √ P² + Q² + 2 PQ cos Θ …………….. ( 2 )

Let β be the angle | R | makes with | P | …. then,

tanβ = NC/ON = NC/OA + AN = Q sin Θ / P + Q cos Θ

β = tan−¹ ( Q sin Θ / P + Q cos Θ ) = direction of the resultant vector.

[ note – analytical treatment of parallelogram method is similar to the above method ]

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** Points to note : **

- If the vectors act in the same direction,i.e, θ = 0° , then R = P+Q as cos 0° = 1 .
- If they are in opposite directions,i.e, θ = 180°, then R = P – Q or Q – P as cos 180° = -1
- If they act at right angles,i.e, θ = 90° , then R = √ P² + Q² as cos 90° = 0 , also note tanβ = Q/P , since sin 90° = 1

**Lami’s theorem :** If the resultant of three vectors is zero, then the magnitude of a vector is directly proportional to the sine of the angle between the other two vectors. *Alternate def* : If the resultant of three vectors is zero,then the ratio of the magnitude of a vector to the sine of the angle between the other two vectors is a constant.

i.e, **P/sinα = Q/sinβ = R/sinγ**

Rectangular Vector Components ( 2 dimensions ) : The component vectors of a vector which is divided into two mutually perpendicular directions in a plane are called rectangular components of the given vector in a plane.

Component along x axis , Ax = A cos Θ ……. horizontal component .. ( 1 )

Component along y axis , Ay = A sin Θ …….. vertical component .. ( 2 )

Now, | A| = | Ax | + | Ay | = Axiˆ + Axjˆ

Squaring and adding equations (1) and (2) , A²x + A²y = A² ( sin²Θ + cos²Θ ) = A²

=> A = √ A²x + A²y and tan Θ = Ay/Ax

Rectangular Vector Components ( 3 dimensions ) : Consider a vector which is split into 3 mutually perpendicular dimensions….

In this case, |A| = |Ax| + | Ay| + |Az| = Axiˆ + Ayjˆ + Azkˆ and

the magnitude of | A | = √ A²x + A²y + A²k

**Directional Cosines :** In the above case**,** r = √ x² + y² + z² and

angles of | r | with x , y, and z axis , respectively are given by

cos α = x/r = l , cos β = y/r = m and cos γ = z/r = n , here, the directional cosines l,m and n of a vector are the cosines of the angles α,β and γ which a given vector makes with x,y and z axis respectively. Now, if we square and add l , m and n

cos ²α + cos² β + cos² γ = x²+y²+z ²/r² or l²+m²+n² = r²/r² = ** 1**

** ** i.e**,** ** the sum of the squares of the directional cosines of a vector is always unity**.

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** Vector Multiplication ( Vector product ) :** Vector multiplication is carried out in two ways…… vector / cross product and scalar / dot product.

Vector / Cross product : Vector product of two vectors | A | and | B | is equal to the product of the magnitudes of | A | and | B | and the sine of the shortest angle between them……. |A| x | B | = AB sinΘ nˆ , where nˆ is the unit vector representing the direction of | A | x | B | and is perpendicular to the plane containing | A | and | B |. The direction of the vector product is the direction a right-hand-thread screw advances.

| A | x | B | = | A x B | = AB sinΘ , also | A | x | B | = | A x B | nˆ

and nˆ = | A | x | B | / | A x B |

Cross product and unit vectors : Now, iˆ.j ˆand kˆ are the unit vectors along the x, y and z axis repectively and the magnitude of each vector is 1 and the angle between any of the two unit vectors is 90° . Hence,iˆxˆj = 1x1xsin 90°nˆ = nˆ . Here, nˆ is a unit vector perpendicular to the plane containing vector i ˆand jˆ.

Rules to note :

- Multiplying any two unit vectors in anti-clockwise direction gives the third unit vector with the positive sign

i.e, iˆ x jˆ= kˆ , jˆx k ˆ= iˆ , kˆ x iˆ = jˆ

- Multiplying any two unit vectors in clockwise direction gives the third unit vector with the negative sign.

** ** jˆx iˆ= – kˆ , kˆx jˆ= – iˆ , iˆxkˆ = – jˆ

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** Cross Product > Component form** : |A| x |B| = (Axiˆ+ Ayjˆ+Azkˆ) x ( Bxiˆ+ Byjˆ+ Bzkˆ)

= AxBx(iˆx iˆ) + AyBx(jˆx iˆ) + Az Bx (kˆx iˆ) + AxBy ( iˆx jˆ) + AyBy( jˆxjˆ ) + AzBy( kˆx jˆ)

+ AxBz ( iˆ x kˆ ) + AyBz( jˆ x kˆ ) + AzBz ( kˆx kˆ )

[ now, iˆxiˆ = 0 , jˆxjˆ = 0 , kˆxkˆ = 0 , and from the above two rules, ]

** |A| x |B| = (AyBz – AzBy) iˆ – (AxBx-AzBx) jˆ + (AxBy – AyBx)kˆ**

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** Cross Product > Determinant method**

** **Here, | A | x | B| = (Axiˆ + Ayjˆ + Azkˆ ) x ( Bxiˆ + Byjˆ + Bzkˆ )

Now,iˆ jˆ and kˆ are used one by one. When iˆ is chosen,it’s corresponding row and column becomes bound and then the remaining elements are subtracted after cross multiplication. Thus, iˆ (AyBz – ByAz) ** -** jˆ (AxBz – BxAz) is the component along iˆ and jˆ . The method is shown below —-

** Cross product / Vector product : Properties**

**Anticommutative :**|A| x |B| = AB sin Θ nˆ and |B| x |A| = BA sin Θ (-nˆ )

i.e, -AB sin Θ nˆ = – ( |A| x |B| ) , hence, |B| x | A| = – |A| x |B| or **|B| x |A| ≠ |A| x |B|**

**Associative :**(|A| + |B|) x (|C| + |D| ) =**|A| x |C| + |A| x |D| + |B| x |C| + |B| x |D|**

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**Distributive : |A| x ( |B| + |C| ) = |A| x |B| + |A| x |C|**

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*Cross / Vector product of two parallel or antiparallel vectors is zero*.**As Θ = 0° , AB sin0°nˆ = o**.*When two vectors are represented by the two adjacent sides of a parallelogram, the magnitude of the cross product will give the area of the parallelogram.*i.e,**AB sinΘ = Ah = base x height of parallelogram = area of parallelogram***When two vectors are represented by the two sides of a triangle,then , half the magnitude of their cross product will give the area of the triangle*. i.e,**ABsinΘ = A x h = base x height = area of the triangle**.*A x ( B + C ) = A x B + A x C – here one product is a scalar and the other is mostly a vector……***why is A x B not useful ? If D = B + C then, AD ≠ AB + AC …. there is no distributive property.****The vector product of any vector with itself is zero.**

**Scalar product ( Dot product )** : Of two vectors | A| and | B | are defined as the product of the magnitudes of the two vectors and the cosine of the smaller angle between them. i.e, |A| x |B| = AB cos Θ

Points to note –

- a . b = b . a i.e, commutative law is followed. In addition, a . ( b+c ) = a.b + a.c [ distributive law ] and λ a.b = λ ( a.b) [ associative with scalar multiplication ].
- a.a = [a ]²
- a.b = 0 , only if a and b are perpendicular or if one of them is zero.
- If [ i,j,k ] is on an orthogonal basis , then i.i = j.j = k.k = 1 and i.j = j.k = k.i = 0
- If a1 = λ1 i + μ1j + ν1k and a2 = λ2i + μ2j + ν2k then a1.a2 = λ1λ2 + μ1μ2 + ν1ν2 .
- A scalar product is a scalar, not a vector and it may be + ve , -ve or zero ——->

- If θ is between 90° and 180°
**———>**cos θ < 0 , hence – ve . - If θ is between 0° and 90° ———> cos θ > 0 , hence + ve .
- When θ = 90° ( vectors are perpendicular ) = 0 .

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**Component of a vector** : Let nˆ be a unit vector. The component of vector ν in the direction of n is defined as νn . If it’s a general vector a , then the component of ν is ν.â.

- if v is a sum of vectors, ν = ν1 +ν2 + ν3, then the component of ν in the direction of n is ν.n = (ν1+ν2+ν3 ) and n = (ν1.n)+(ν2.n)+(ν3.n)and this isbased on the distributive law for a scalar product. Therefore, the component of the sum of a no of vectors in a given direction is equal to the sum of the components of the individual vectors in that direction.
- if a vector ν is expanded in terms of a general basis set ( a,b,c ) in the form λa + μb + νc, the co-efficients λ,μ and ν are not components of vector ν in the direction of a,b and c. But , if ν is expanded in terms of an orthonormal basis set [ i,j,k ] in the form ν = λi +μj + νk, then the component of ν in the i direction is ν. i = ( λi +μj + νk ) . i = λ( i.i ) +μ( j. i ) + ν( k.i ) = λ + 0+0 = λ . Likewise, μ and ν are the components of ν in the j and k directions.
**Therefore**, when a vector ν is expanded in terms of an orthonormal basis set ( i ,j , k ) in the form ν =λi +μj + νk , the co-efficients λ , μ and ν are the components of ν in the i , j and k directions.

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**Method of Components :** Adding vectors by measuring a scale diagram offers limited accuracy and calculations with right triangles work only when the two vectors are perpendicular.So.another method is to add the components of a vector i.e, | A | = | Ax | + | Ay |.*Components* *of** vectors are not vectors themselves,they are just numbers.*The components of a vector may be

**+ve or -ve numbers**.

** Note** : Relating a vector’s magnitude and direction to it’s components **are correct only when** the angle θ is measured from the +ve x axis .When finding the direction of a vector from it’s components, check to which quadrant the angle belongs to.Eg,if tan θ = -1 , the angle could be either 135° or 315°…….. hence,only by checking the components, the angle can be found out.

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** VECTOR CLASSIFICATION** : Based on the character of their magnitude / direction or both,vectors can be broadly classified as :-

: Vectors having a starting point or point of application.*Polar vectors ( true vectors )*(*Axial vectors*) : Vectors whose directions are along the axis of rotation. Called pseudo vector because*pseudo vectors**it’s sign**changes when the orientation in space changes*. Such a sign change doesn’t happen in a polar vector.Pseudo vectors usually occur as the cross – product of two normal vectors. Eg : Angular velocity, Torque, Magnetic feild.: Two or more vectors acting along the same line or along the parallel lines. They may act either in the same direction or in the opposite direction.*Collinear vectors*: Collinear vectors having the same direction . Hence, angle between them is 0°.Consider two vectors, |A| and |B| which are parallel.Here, |B| can be written as m|A| ,where m is a number.For parallel vectors m is +ve and for antiparallel it is -ve.For parallel vectors, their unit vectors are equal. i.e, Aˆ = Bˆ and if antiparallel Aˆ = -Bˆ*Parallel vectors*: Collinear vectors in opposite direction .Hence,angle between them is 180° or Π radian.*Anti-parallel vectors*: Two vectors having equal magnitude and same direction.*Equal vectors*: Vectors lying in the same plane.Consider three vectors, |A| , |B| and |C| which are coplanar.In this case,|A| can be written as |A| = m|B| + n |C|, where m and n are numbers.*Coplanar vectors*If we take four vectors in any direction or plane we choose,they can be written as |A| = m|B| + n|C| + o|D| , where m , n and o are some numbers.*The minimum number of unequal vectors whose sum can be zero is three and those three vectors should be coplanar.The minimum no of non-coplanar vectors whose sum can be zero is**four*.: Vector which provides an idea about the direction and distance of a point from origin in space.*Position vector ( Radius vector )*: A vector whose length is zero. In co-ordinates , the vector is (0,0,0), and it is commonly denoted , or*Null vector ( zero vector )***0.**It doesn’t have a direction and cannot be**normalized..**i.e**,**a unit vector is not possible,which is a multiple of a null vector. The sum of the null vector with any vector**a**is**a**(that is,**0**+**a**=**a**).: Resultant vector of two or more vectors is defined as that single vector which produces the same effect both in magnitude and direction as produced by individual vectors taken together. [ Here resultant refers to addition ]*Resultant vector*

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**S****calar triple product** ( **box product** or **mixed triple product )** : Refers to a method of applying the vector product or scalar product to three vectors. It could be denoted as (**a** **b** **c**)

** ( a b c ) = a . ( b x c )**

Points to consider :

- Used in finding the volume of the parallelepiped which has edges defined by three vectors .
- It’s product could be zero, if all the vectors lie in the same plane ( linearily dependent ),and hence can’t make a volume.
- It’s value is +ve , if all the three vectors are right handed.