*Vectors and ScalarsA change in the position of particle is called displacement, e.g. when a particle moves from position A to B, we represents its displacement by drawing a line from A to B with an arrowhead at B representing the direction of motion of particle. The straight line from A to B with an arrowhead at B represents the net effect of the motion of particle not the actual path and is called displacement. A displacement is therefore characterized by a length and a direction. Physical quantities that needs magnitude and direction for their complete description are called vectors, similar other quantities that are vectors; are force, velocity, acceleration etc.Vector representation: Quantities that can be described only by magnitude without any direction are called scalars, These quantities therefore have only a number and unit which describe their magnitude and there is no direction, such as time, mass, speed, etc.

*Addition of vectors by graphical methodFollowing rules should be followed for addition of vectors Make a proper scale Draw vector a in a proper direction in the co ordinate system Draw vector b to the same scale with its tail at the head of a and with its own proper direction. Draw a line from the tail of a to the head of b to represent the vectors sum s. s is called the resultant vector and has the equivalent effect of the combining effect of a and b.If a and b are displacements then s will represents the net effect of these displacement

*Properties of vectors additionCommutative law: This law assert that it will make no difference in what order we are adding vectors.Associative law:This law assert that it makes no difference in what group we add vectors.Vectors subtraction: By making the negative of a vector to be subtracted from an other vector and then adding with that; will perform the subtraction operation of vectors, where negative of a vector is the vector which has the same magnitude as the original but has opposite direction

*Components of vectorsLet we have vector a in xy co ordinate system with its tail at the origin O of the system, and let draw perpendiculars from the head of vector a to the axes, then the quantities ax and ay are called the Cartesian components of vector a and the process is called resolution of vector into its components. The vector a is completely and uniquely described by these components i.e. ax and ay and would reconstruct from these components.The components of vector can be positive or negative, or zero. The components of vector a along X and Y axes are given as under and are called rectangular components, where is the angle which a makes along x-axesax = a cos, ay = a sin

*The sign of the components of a vector depends upon the quadrant in which the angle lies, e.g. the vector always has a negative x-components, when lies between 90o and 180o.Here a represents the magnitude of the vector and indicates its direction relative to x- axes.The components of vector ax and ay themselves can be used to specify the vector. From the figure on previous slide we now thatThe quadrant in which lies specify the sign of ax and ay

*Components of vectors in three dimensionIn order to find the components of vector along three directions (x, y, z), make a projection of vector a in the xy-plane and from the tip of that projection draw the components ax and ay.ax = a sin cos, ay = a sin sinaz = a cosThe angle between vector a and z- axes is called polar angle, and the angle in the xy plane between the projection of a and axes is called azimuthal angle

*Unit vectors in three dimensionUsually unit vectors are introduce with the components of vector along the three directions. These vectors has a magnitude of unity and are useful for the determination of a particular direction.In the rectangular coordinate system the special symbols i, j, and k are usually used for unit vectors in the positive x, y, and z directions respectively.In general a vector a in a three dimensional coordinate can be written in its components form as undera = axi + ayj + azkComponents of vector in two dimension

*Addition of vectors by component methodThere are some rules for adding vectors by this method and these rules must be followed when adding vectors. Resolve each vector into its components Add the components of various vectors for each coordinate axes i.e. x, y and z axes. The sum so obtained are the components of the resultant vectors Construct the resultant vector for the components of resultant vectorsLet we have two vectors a and b and s is the sum of these two vectors, then we have s = a + b (1)Two vectors are equal to each other only if their corresponding components are also equal to each other.

*This means that the components of s must be equal to the components of a + b or in other wards sxi +syj = (axi +ayj) + (bxi + byj)sxi +syj = (axi + bxi) + (ayj + byj) (2)Equating the components on both sides of equation 2sx = ax + bxsy = ay + by(x- components of the resultant vector)(y- components of the resultant vector)The magnitude and direction of resultant vector s can be given as s = (s2x + s2y)1/2 (3)s = [(ax +bx)2 + (ay + by)2]1/2 (4)tan = sy/sx = (ay + by)/(ax + bx) (5)

*Vector multiplicationLike scalars, vectors can also be multiplied to obtain an other physical quantity of new dimension, because vectors have magnitude as well as direction. We use some rules for vectors multiplication, and there are three types of vectors multiplication given as under. Multiplication of vector by a scalar.The multiplication of vector by a scalar is very simple, it is actually the multiplication of vector a by a scalar quantity c, if c is positive then the product will have the same direction as a but the magnitude will be c time a, and when c is negative then the product will have direction opposite to a. a-1.5a2a

*Multiplication of two vectors to give scalarWhen the multiplication of two vectors is a scalar quantity, then it is called scalar product, and can be given as a.b = ab cos (6)Where a is the magnitude of a and b is the magnitude of b and cos is the cosine of angle between a and b. because of the notation, this type of vectors product is also called dot product. In the above equation a, b and cos are all scalars. The scalar product of the two vectors is actually the product of the magnitude of one vector with the component of the other vector in the direction of first vector. a.b = a (b cos)a.b = b (a cos)

*The scalar product of two vectors that are perpendicular to each other vanishes. Therefore the dot product of three Cartesian unit vectors is always zeroi.j = j.k = k.i = 0 (7)i.i = j.j= k.k = 1 (8)The dot product of unit vector with itself gives unityThe dot product of two vectors a and b in a three dimension coordinate system can be given asa.b = axbx +ayby +azbz (9)

*Multiplication of two vectors to give vectorWhen the multiplication of two vectors give an other vector, this type of product is called vector product or cross product, because there is a cross between the two vectors.Let we have two vectors a and b their vector product will be an other vector c whose direction will be perpendicular to the plane containing a and b. where the magnitude of c is defined by the following equationFor the direction of vector c use right hand rules, as shown in the figure

*Draw the vectors a and b intersecting their tails at the origin and consider an axis perpendicular to the plane containing a and b. Wrap the fingers of the right hand around this axis and push the vector a (first in the product) into the vector b (second in the product) with the smaller angle; the direction of the thumb gives the direction of the vector product a x b or c. When the angle between the vectors is 90o then a, b, and c are all at right angle to each other. The vector product of two perpendicular vectors such as unit vectors i, j, and k give an other unit vector that is perpendicular to the previous two.i x j = k, j x k = i, k x i = jThe order of factors in vector product is of most importance, because the product b x a is not the same as a x b but a x b = - b x a

* similarly the cross product of any two( like unit vectors) also vanishes. i x i = j x j = k x k = 0The cross product of any vector with itself vanishes i.e.a x a = 0The cross product of any two vectors in three dimension can be given as = (aybz azby)i + (azbx axbz)j + (axby - aybx)kWhere a = axi + ayj +azk b = bxi + byj + bzk

*