SCALAR AND VECTOR
description
Transcript of SCALAR AND VECTOR
By:
Engr. Hinesh KumarLecturer
I.B.T, LUMHS, Jamshoro
Scalars are quantities which have magnitude without direction.
Examples of scalars
• temperature• mass• kinetic energy
• time• amount• density• charge
Scalars
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
Contd….The symbol for a vector is a letter with an
arrow over it.
All vectors have head and tail.
A
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
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.
y
x
AAx
AyA
The direction of A can be stated as
tan = Ay / Ax
=tan-1(Ay / Ax)
Vector Representation of ForceForce has both magnitude and direction and
therefore can be represented as a vector.
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.
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
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
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.
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
x1
x5
x4
x3
x2
xi
xi = x1 + x2 + x3 + x4 + x5
ExampleExample
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
Vector SubtractionEquivalent to adding the negative vector
A
-BA - B
B
A BC =
A + (-B)C =
ExampleExample
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
B = A
If = 0, therefore B = A = 0, which is also known as a zero vector
(A) = A = (A)
(+)A = A + A
ExampleExample
Rules of Vector Addition commutative
A + B = B + A
A
B
A + BB
A A + B
associative
(A + B) + C = A + (B + C)
B
CA
B
CA A + B
(A + B) + CA + (B + C)
B + C
distributive
m(A + B) = mA + mB
A
B
A + B mA
mB
m(A + B)
Parallelogram method of addition (tailtotail)
A
B
A + B
The magnitude of the resultant depends on the relative directions of the vectors
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
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
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
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
The unit vector notation for the vector AA is written
A = Axi + Ayj
x- axis
y- axis
Ax
Ay
θ
A
i
j
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
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
Dot product (scalar) of two vectors
The definition:
θ
B
AA · B = │A││B │cos θ
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
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 .
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
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
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
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
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
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
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
(1, 0)
(2, 2)
x1 + x2 = (1, 0) + (2, 2)= (3, 2)
x1
x2
x1 + x2
(1, 0)
(2, 2)
(x2)
x1
x1 + x2x2
x1 + x2 = (1, 0) + (2, 2)= (3, 2)
x1 - x2?
(1, 0)
(2, 2)
x1
-x2x1 - x2
x1 - x2 = (1, 0) - (2, 2)= (-1, -2)
x1 - x2 = x1 + (-x2)
Example -2D for subtraction
(1, 0)
(2, 2)
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