Vector Analysis (1)
Sung Joon Maeng
School of Electrical Engineering
Hanyang University ERICA Campus
Scalars and Vectors
▪ Scalar
▪ A single positive or negative real number
▪ Example:
▪ Distance, temperature, time
▪ 0, -1, 1, 3.14
▪ Vector
▪ Magnitude and direction (in a space)
▪ Focus of this course
▪ Two or three-dimensional spaces
▪ Generally, vectors can have n-dimensional space
▪ Example: Force, velocity
▪ Boldface type: A, a
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Scalars and Vectors
▪ Field
▪ Definition:
▪ A spatial distribution of a quantity, which may or may not be a function
of time.
▪ The field concept invariably is related to a region.
▪ Scalar and vector fields
▪ Scalar field (example)
▪ Temperature throughout a bowl of soup
▪ Density at any point in the earth
▪ Vector field (example)
▪ Gravitational field of the earth
▪ Magnetic field of the earth
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Vector Addition
▪ Parallelogram law
▪ Commutative law
▪ A+B=B+A
▪ Associative law
▪ A + (B + C) = (A + B) + C
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Vector Subtraction
▪ Parallelogram law
▪ From addition
▪ A – B = A + (-B)
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Multiplication and Division
▪ Multiplication by scalars
▪ Magnitude: scales by the scalar value
▪ Direction: for a positive scalar, no change on the direction.
▪ Associative and distributive laws
(r + s) (A + B) = r (A + B) + s (A + B) = r A + r B + s A + s B
▪ Division by scalars
▪ Multiplication by the reciprocal of those scalars
A 1
= A
r r
▪ Multiplication by vectors?
▪ Dot product and Cross product
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Rectangular Coordinate System
▪ Coordinate systems
▪ Rectangular coordinates
▪ Circular cylindrical coordinates
▪ Spherical coordinates
▪ Rectangular coordinate system
▪ Three coordinate axes: x, y, z
▪ Right-handed coordinate system
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Point Locations in Rectangular Coordinates
▪ A point is located by giving its x, y, and z coordinates
▪ x (y, or z) shows the distance from the point to the plane with x=0.
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Differential Volume Element
▪
▪
▪
▪
▪
▪
Point P with (x, y, z)
Increase each coordinate by a differential amount
P’ with (x+dx, y+dy, z+dz)
The six planes related to P and P’ consists a parallelepiped
Differential volume: dxdydz
Differential area: dxdy, dydz and dzdx
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Summary
• Rectangular coordinates
• Point representation
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Orthogonal Vector Components
▪ Purpose
▪ Describe a vector in the rectangular coordinate system.
▪ Logical way
▪ Three component vectors along the three coordinate axes (x, y and z.)
▪ r=x+y+z
▪ x, y, z are the component vectors.
▪ x, y, z have magnitudes depends on
the given vector r
but a known constant direction.
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Orthogonal Unit Vectors
▪ Unit vectors
▪ Having unit magnitude
▪ Parallel to the coordinate axes
▪ Pointing in the direction of increasing coordinate values
▪ Rectangular coordinate system
▪ ax, ay and az
▪ Directing the x, y and z axes
▪ P(1, 2, 3)
▪ rP = ax + 2ay + 3az
▪ 1, 2, 3 are components
▪ ax , 2ay , 3az are component vectors
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Vector Representation
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Summary
• Vector representation
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Vector Expressions in Rectangular Coordinates
General vector B:
Magnitude of B:
Unit vector in the
direction of B:
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Example 1.1
▪ Specify the unit vector extending from the origin
toward the point G(2, -2, -1)
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Example D1.1
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Vector Field
▪ We are accustomed to thinking of a specific vector
▪ A vector field is a function defined in space that has
magnitude and direction at all points:
where r = (x, y, z) represents the position vector
Each of the components vx, vy, and vz may be a function of the x, y, and z
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Example D1.2
▪ Given a vector field S which is expressed in a
rectangular coordinates as
▪ Specify the vector field at P(2, 4, 3)
▪ Specify the surface f (x,y,z) on which |S|=1
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Dot Product
▪ Definition
▪ For two vectors A and B
▪ The product of the magnitude of A, the magnitude of B, and the
cosine of the smaller angle between them.
Scalar
▪ Commutative law:
▪ Distributive law
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Dot Product
Given
Find
Distributive law and
We have
Note also:
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Vector Projections using the Dot Product
a: unit vector
B • a gives the scalar component of B
in the horizontal direction
(B • a)a gives the vector component
of B in the horizontal direction
The geometrical term projection is also used with the dot product.
B • a is the projection of B in the a direction.
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Example 1.2
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Example D1.3
The three vertices of a triangle are located at A(6,−1, 2), B(−2, 3,−4),
and C(−3, 1, 5). Find:
(a) RAB
(b) RAC
(c) the angle θBAC at vertex A
(d) the (vector) projection of RAB on RAC .
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Cross Product
▪ Definition
▪ For two vectors A and B
▪ Magnitude: the product of the magnitude of A, the magnitude of B,
and the sine of the smaller angle between them.
▪ Direction: perpendicular to the plane including A and B, (righthanded rule)
Dot product: scalar product
Cross product: vector product
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Cross Product
▪ The cross product is not commutative because
▪ Distributive law
▪ Right-handed rectangular coordinate system
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Cross Product in Rectangular Coordinates
Begin with:
where
Therefore:
Or…
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Example
▪
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Example D1.4
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