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Chapter one Vector Analysis. A scalar quantity is one which can be described fully by just stating its magnitude . Some examples are Mass time length temperature density speed energy and volume.

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Chapter one Vector Analysis

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Chapter one vector analysis

Chapter one

Vector Analysis


Scalars vectors

A scalar quantity is one which can be described fully by just stating its magnitude.

Some examples are

Mass

time

length

temperature

density

speed

energy and volume

A vector quantity is one which can only be fully described if its magnitude and direction stated.

Some examples are

displacement

velocity

acceleration

force

momentum

magnetic density and electric intensity.

ScalarsVectors


Why vectors are important

Why vectors are important?

  • Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0,5). Another quantity represented by a vector is force, since it has a magnitude and direction.


Properties of a vector

Properties of a Vector

  • A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol A.

  • The magnitude of A is |A| ≡A

  • We can represent vectors as geometric objects using arrows. The length of the arrow corresponds to the magnitude of the vector. The arrow points in the direction of the vector

A

|A| ≡A

O


Types of vector

Types of Vector

  • Null Vector

    • Vector with zero magnitude

  • Position Vector

    • Vector starting from origin

  • Free Vector

    • Vector starting from anywhere but origin

  • Unit Vector

    • Vector with magnitude of 1

  • Equal Vectors

    • Two vectors equal in magnitude and direction

  • Opposite Vectors

    • Two vectors equal in magnitude but opposite in direction.


  • Laws of vector algebra

    Laws of vector algebra

    • A+B=B+A(Commutative law of addition)

    • A+(B+C)=(A+B)+C (Associative law of addition)

    • mA=Am (Commutative law of Multiplication)

    • m(nA)=(mn)A (Associative law of Multiplication)

    • (m+n)A=mA+nA(Distributive law)

    • m(A+B)=mA+mB (Distributive law)


    Addition of vectors

    Addition of Vectors

    • The addition of two vectors yields another vector known as Resultant vector.

    • For example if vector A and vector B are added their sum will be equal to (A+B).


    Methods of addition of vectors

    Methods of Addition of Vectors

    There are 3 methods for addition of Vectors.

    • Addition Algebraically

      • For vectors precisely along X or Y axis.

  • Parallelogram law of addition

  • Triangle law of addition.


  • Parallelogram law

    Parallelogram law

    • Vector P and Q are drawn from same origin.

    • Straight lines are drawn parallel to both vectors so as to form a parallelogram.

    • The resultant (P+Q) is represented by the diagonal of the parallelogram that passes through the origin.

    +

    Q

    P

    Q

    +

    Q

    P

    P


    Triangle law head to tail rule

    Triangle law (head to tail rule)

    • The result of two vectors could be determined by drawing a triangle.

    • Vector Q and P are drawn in such a way that the tail of vector Q touches head of vector P.

    • The resultant (P+Q) is represented by the third side of triangle from tail of P to head of Q.

    +

    P

    Q

    Q

    +

    P

    Q

    P


    Subtraction of vectors

    Subtraction of vectors

    • The subtraction of two vectors can be treated as the addition of a negative vector.

      (P-Q)=P+(-Q)

    • The vector (P-Q) can then be determined by any of the two methods.

    -

    P

    Q

    =

    P

    -Q

    P

    -Q

    OR

    P

    P-Q

    -Q


    Resolution of a vector

    Resolution of a vector

    • Vector R could be considered to be the resultant of two vectors.

      R=A+B

    • Here the vectors A and B are known as the components of vectors.


    Resolution of a vector1

    Resolution of a vector

    • It is useful to find the components of a vector R in two mutually perpendicular directions. This process is known as resolving a vector into components.

    • The magnitude of the two components can be written in the form Rcosy and Rsiny

    R

    Rsin0

    0

    Rcos0

    Rsind

    R

    d

    Rcos d


    Triangle of forces

    Triangle of forces

    • If three forces acting on a point can be represented in magnitude and direction by three sides of a triangle taken in order, then the three forces are in equilibrium.

    • The converse is also true:

      • Three forces acting on a point are in equilibrium, they can be represented in magnitude and direction by sides of a triangle taken in order.


    Triangle of forces1

    Triangle of forces

    Q

    • By the triangle of vectors, the resultant of P and Q is represented in magnitude and direction by side of OC of the triangle OAC.If third force R is equal in magnitude to (P+Q) but in opposite direction, then the point O is in equilibrium and also R could be represented by the side CO of the triangle.

    C

    A

    P

    Q

    P

    P+Q

    o

    O

    R

    R

    Q

    C

    A

    P

    R

    O


    Polygon of forces

    Polygon of Forces

    • When more than three coplanar forces act on a point , the resultant (or vector sum) of forces can be found by drawing a polygon of forces.


    Polygon of forces1

    Polygon of Forces

    S

    • If forces acting on a point can be represented in magnitude and direction by sides of a polygon taken in order ,then the forces are in equilibrium.

    T

    o

    P

    Q

    R

    D

    R

    S

    E

    C

    T

    Q

    P

    A

    B


    Resultant of a number of forces

    Resultant of a number of forces

    • If point O is being acted upon by a number of coplanar forces such as A,B,C,D,E do not form a closed polygon then the forces are not in equilibrium

    • The resultant in magnitude and direction is represented by R.

    B

    D

    A

    C

    o

    E

    D

    C

    E

    B

    R

    A


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