SCALAR AND VECTOR
description
Transcript of SCALAR AND VECTOR
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By:
Engr. Hinesh KumarLecturer
I.B.T, LUMHS, Jamshoro
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Scalars are quantities which have magnitude without direction.
Examples of scalars
• temperature• mass• kinetic energy
• time• amount• density• charge
Scalars
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VectorA vector is a quantity that has both magnitude (size) and direction.
it is represented by an arrow whereby– the length of the arrow is the magnitude, and– the arrow itself indicates the direction
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Contd….The symbol for a vector is a letter with an
arrow over it.
All vectors have head and tail.
A
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Two ways to specify a vector
It is either given by• a magnitude A, and• a direction
Or it is given in the x and y components as
• Ax
• Ay
y
x
A
A
Ay
x
Ax
Ay
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y
x
AAx
AyA
Ax = A cos
Ay = A sin
│A │ =√ ( Ax2
+ Ay2
)
The magnitude (length) of A is found by using the Pythagorean Theorem
The length of a vector clearly does not depend on its direction.
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y
x
AAx
AyA
The direction of A can be stated as
tan = Ay / Ax
=tan-1(Ay / Ax)
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Vector Representation of ForceForce has both magnitude and direction and
therefore can be represented as a vector.
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Vector Representation of Force
The figure on the left shows 2 forces in the same direction therefore the forces add. The figure on the right shows the man pulling in the opposite direction as the cart and forces are subtracted.
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Some Properties of Vectors
Equality of Two Vectors
Two vectors A and B may be defined to be equal if they have the same magnitude and point in the same directions. i.e. A = B
A BA
A
B
B
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Negative of a VectorThe negative of vector A is defined as giving the vector sum of zero value when added to A . That is, A + (- A) = 0. The vector A and –A have the same magnitude but are in opposite directions.
A
-A
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Applications of VectorsVECTOR ADDITION – If 2 similar vectors point in
the SAME direction, add them.
Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started?
54.5 m, E 30 m, E+
84.5 m, E
Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION.
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The addition of two vectors A and B
- will result in a third vector C called the resultant
C = A + B
A
BC
Geometrically (triangle method of addition)
• put the tail-end of B at the top-end of A• C connects the tail-end of A to the top-end of B
We can arrange the vectors as we like, as long as we maintain their length and direction
Vector Addition
ExampleExample
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x1
x5
x4
x3
x2
xi
xi = x1 + x2 + x3 + x4 + x5
ExampleExample
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Applications of VectorsVECTOR SUBTRACTION - If 2 vectors are going
in opposite directions, you SUBTRACT.
Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started?
54.5 m, E
30 m, W-
24.5 m, E
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Vector SubtractionEquivalent to adding the negative vector
A
-BA - B
B
A BC =
A + (-B)C =
ExampleExample
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Scalar MultiplicationThe multiplication of a vector Aby a scalar
- will result in a vector B
B = A- whereby the magnitude is changed but not the direction
• Do flip the direction if is negative
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B = A
If = 0, therefore B = A = 0, which is also known as a zero vector
(A) = A = (A)
(+)A = A + A
ExampleExample
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Rules of Vector Addition commutative
A + B = B + A
A
B
A + BB
A A + B
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associative
(A + B) + C = A + (B + C)
B
CA
B
CA A + B
(A + B) + CA + (B + C)
B + C
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distributive
m(A + B) = mA + mB
A
B
A + B mA
mB
m(A + B)
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Parallelogram method of addition (tailtotail)
A
B
A + B
The magnitude of the resultant depends on the relative directions of the vectors
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a vector whose magnitude is 1 and dimensionless
the magnitude of each unit vector equals a unity; that is, │ │= │ │= │ │= 1
i a unit vector pointing in the x direction
j a unit vector pointing in the y direction
k a unit vector pointing in the z direction
and defined as
Unit Vectors
k
j
i
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Useful examples for the Cartesian unit vectors [ i, j, ki, j, k ] - they point in the direction of the x, y and z axes respectively
x
y
z
ii
jj
kk
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Component of a Vector in 2-D vector A can be resolved into two
components Ax and Ay
x- axis
y- axis
Ay
Ax
A
θ
A = Ax + Ay
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The component of A are
│Ax│ = Ax = A cos θ
│Ay│ = Ay = A sin θ
The magnitude of A
A = √Ax2 + Ay
2
tan = Ay / Ax
=tan-1(Ay / Ax)
The direction of A
x- axis
y- axis
Ay
Ax
A
θ
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The unit vector notation for the vector AA is written
A = Axi + Ayj
x- axis
y- axis
Ax
Ay
θ
A
i
j
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Component of a Unit Vector in 3-D vector A can be resolved into three
components Ax , Ay and Az
A
Ax
Ay
Az
y- axis
x- axis
z- axis
i
j
k
A = Axi + Ayj + Azk
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if
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
A + B = C sum of the vectors A and B can then be obtained as vector C
C = (Axi + Ayj + Azk) + (Bxi + Byj + Bzk)
C = (Ax + Bx)i+ (Ay + By)j + (Az + Bz)kC = Cxi + Cyj + Czk
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Dot product (scalar) of two vectors
The definition:
θ
B
AA · B = │A││B │cos θ
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if θ = 900 (normal vectors) then the dot product is zero
Dot product (scalar product) properties:
if θ = 00 (parallel vectors) it gets its maximum
value of 1
and i · j = j · k = i · k = 0|A · B| = AB cos 90 = 0
|A · B| = AB cos 0 = 1 and i · j = j · k = i · k = 1
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A + B = B + A
the dot product is commutative
Use the distributive law to evaluate the dot product
if the components are known
A · B = (Axi + Ayj + Azk) · (Bxi + Byj + Bzk)
A. B = (AxBx) i.i + (AyBy) j.j + (AzBz) k.k
A .
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Cross product (vector) of two vectorsThe magnitude of the cross product given by
the vector product creates a new vector
this vector is normal to the plane defined by the
original vectors and its direction is found by using the
right hand rule
│C │= │A x B│ = │A││B │sin θ
θ
A
BC
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if θ = 00 (parallel vectors) then the cross
product is zero
Cross product (vector product) properties:
if θ = 900 (normal vectors) it gets its maximum
value
and i x i = j x j = k x k = 0|A x B| = AB sin 0 = 0
|A x B| = AB sin 90 = 1 and i x i = j x j = k x k = 1
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the relationship between vectors i , j and k can
be described as
i x j = - j x i = k
j x k = - k x j = i
k x i = - i x k = j
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Example 1 (2 Dimension)
If the magnitude of vector A and B are equal to 2 cm and 3 cm respectively , determine the magnitude and direction of the resultant vector, C for
B
Aa) A + B
b) 2A + B
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Solution
a) |A + B| = √A2 + B2
= √22 + 32
= 3.6 cm
The vector direction
tan θ = B / A
θ = 56.3
b) |2A + B| = √(2A)2 + B2
= √42 + 32
= 5.0 cm
The vector direction
tan θ = B / 2A
θ = 36.9
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Example 2
Find the sum of two vectors A and B lying in the xy plane and given by
A = 2.0i + 2.0j and B = 2.0i – 4.0j
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SolutionComparing the above expression for A with the general relation A = Axi + Ayj , we see that Ax= 2.0 and Ay= 2.0. Likewise, Bx= 2.0, and By= -4.0 Therefore, the resultant vector C is obtained by using EquationC = A + B + (2.0 + 2.0)i + (2.0 - 4.0)j = 4.0i -2.0j
or Cx = 4.0 Cy = -2.0
The magnitude of C given by equation
C = √Cx2 + Cy
2 = √20 = 4.5
Find the angle θ that C makes with the positive x axis
Exercise
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(1, 0)
(2, 2)
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x1 + x2 = (1, 0) + (2, 2)= (3, 2)
x1
x2
x1 + x2
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(1, 0)
(2, 2)
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(x2)
x1
x1 + x2x2
x1 + x2 = (1, 0) + (2, 2)= (3, 2)
x1 - x2?
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(1, 0)
(2, 2)
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x1
-x2x1 - x2
x1 - x2 = (1, 0) - (2, 2)= (-1, -2)
x1 - x2 = x1 + (-x2)
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Example -2D for subtraction
(1, 0)
(2, 2)
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AssignmentIf one component of a vector is not zero, can its magnitude be zero? Explain and Prove it.
If A + B = 0, what can you say about the components of the two vectors?
A particle undergoes three consecutive displacements d1 = (1.5i + 3.0j – 1.2k) cm,
d2 = (2.3i – 1.4j – 3.6k) cm d3 = (-1.3i + 1.5j) cm. Find the component and its magnitude.
1
2
3
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