Vectors

What is Vector? In elementary mathematics, physics, and engineering, a

Euclidean vector is a geometric object that has both a magnitude (or length) and direction.

A mathematical expression possessing magnitude and direction, which add according to parallelogram law

In statics the vector quantities frequently encountered are position, force and moment.

Vectors

Magnitude of a vector is given by its length. The magnitude is designated as IABI or simply AB. It is always a positive quantity, is symbolized in italic type.

The direction is defined by the angle between a refrence axis and the arrow’s line of action and the sence is indicated by the arrowahead.

For example; The vector shown in fig has a

magnitude of 20 miles, a direction which is measured 60 degrees counter clockwise.

Vectors-Scalar Multiplication

Scalar Multiplication of Vector: The product of vector A and a scalar b, yielding bA, is

defined as a vector having a magnitude IbAI.

The sense of bA is the same as A provided b is positive, it is opposite to A if b is negative.

Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.

Vectors-Scalar Multiplication

Fig i

Fig ii

Vectors-Scalar Multiplication

Properties of Scalar Multiplication of Vector:

Scalar multiplication obeys the following rules (vector in boldface):

Left distributive: (c + d)v = cv + dv; Right distributive: c(v + w) = cv + cw; Associative: (cd)v = c(dv); Multiplying by 1 does not change a vector: 1v = v; Multiplying by 0 gives the null vector: 0v = 0; Multiplying by -1 gives the additive inverse: (-1)v = -v.

Vectors-Scalar Division

Scalar Division of Vector: Scalar division is performed by multiplying the vector

operand with the numeric inverse of the scalar operand. This can be represented by the use of the fraction bar or division signs as operators. The quotient of a vector v and a scalar c can be represented in any of the following forms:

Vectors

Norm The norm of a vector is represented with double bars on

both sides of the vector. The norm of a vector v can be represented as:

IIvII The norm is also sometimes represented with single bars,

like IvI, but this can be confused with absolute value (which is a type of norm).

Vectors-Addition

Vector Addition:

Two or more vector can be added according to parallelogram law.

Parallelogram Law Of Vector Addition:

"If two vector quantities are represented by two adjacent sides or a parallelogramthen the diagonal of parallelogram will be equal to the resultant of these two vectors."

Vectors-Addition

Resultant Vector:

The sum of two or more vectors can be written as another vector, which is called the resultant vector. It is the result of adding vectors together.

Resultant Force:In mechanics, a single force acting on a particle or body whose effect is equivalent to the combined effects of two or more separate forces. The resultant of two forces acting at one point on an object can be found using the parallelogram of forces method

Vectors-Addition

Example:Consider two vectors . Let the vectors have the following orientation fig (i)

Parallelogram of these vectors is shown in fig (ii) According to parallelogram law:

fig i fig ii

Vectors-Addition

Magnitude Of Resultant Vector: Magnitude or resultant vector can be determined by using

either sine law or cosine law.

Vectors-Subtraction

Vector Subtraction:

It is defined as the addition of corresponding negative vectors.

Commutative law:

Since the parallelogram constructed on both the vectors does not depends upon the order in which are selected, we conclude that the addition and subtraction of two vectors is commutative;

Vectors-Addition

Triangle Rule The procedure of "the triangle of vectors addition method" is

Draw vector A using appropriate scale and in the direction of its action

From the nose of the vector draw vector B using the same scale and in the direction of its action

The resulting vector R is represented in both magnitude and direction by the vector drawn from the tail of vector A to the nose of vector B

Vectors-Laws

Associative Law:The operations of addition and subtraction are associative, illustrated by following equation;

Unit Vector:When a vector is multiplied by the reciprocal of its magnitude, the resultant vector is know as unit vector in the direction of the original vector. Relationship is expressed by the equation:

Vectors-Polygon Rule

Polygon Rule:A particle is acted upon by several coplanar i.e. by several forces in the same plane. The resultant can be found by polygon rule. Use of polygon is equivalent to repeated application of parallelogram law, as shown in figure below;

Vectors-Polygon Rule

If the polygon is closed, the resultant is a vector of zero magnitude and has no direction. This is called the null vector.

Null Vector:It is vector whose components are zero. Also know as zero vector or empty vector.

Vectors-Resolution

RESOLUTION OF VECTORThe process of splitting a vector into various parts or components is called "Resolution Of Vector“ These parts of a vector may act in different directions and are called "components of vector". We can resolve a vector into a number of components .Generally there are three components of vector viz. Component along X-axis called x-componentComponent along Y-axis called Y-componentComponent along Z-axis called Z-component Here we will discuss only two components x-component & Y-component which are perpendicular to each other. These components are called rectangular components of vector.

Vectors-Resolution

Method Of Resolving A Vector Into Rectangular ComponentsConsider a vector acting at a point making an angle with positive X-axis. Vector is represented by a line OA. From point A draw a perpendicular AB on X-axis. Suppose OB and BA represents two vectors. Vector OA is parallel to X-axis and vector BA is parallel to Y-axis. Magnitudeof these vectors are Vx and Vy respectively. By the method of head to tail we notice that the sum of these vectors is equal to vector .Thus Vx and Vy are the rectangular components of vector .

Vectors-Resolution

Vx = Horizontal component of .Vy = Vertical component of .

Vectors-Resolution

Magnitude Of Horizontal Component

Consider right angled triangle OAB

Magnitude Of Vertical Component

Consider right angled triangle OAB

Materials-Iron

Lecture Recap

http://www.youtube.com/watch?v=QQz0TiV4vJY&feature=player_detailpage

http://www.youtube.com/watch?v=fOasNol5iD0&feature=player_detailpage