Scalar and Vector Fields A scalar field is a function that gives us a single value of some variable...
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Scalar and Vector Fields A scalar field is a function that gives us a single value of some variable for every point in space. Examples: voltage, current,
Scalar and Vector Fields A scalar field is a function that
gives us a single value of some variable for every point in space.
Examples: voltage, current, energy, temperature A vector is a
quantity which has both a magnitude and a direction in space.
Examples: velocity, momentum, acceleration and force
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Example of a Scalar Field
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3 Scalar Fields e.g. Temperature: Every location has associated
value (number with units)
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4 Scalar Fields - Contours Colors represent surface temperature
Contour lines show constant temperatures
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5 Fields are 3D T = T(x,y,z) Hard to visualize Work in 2D
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6 Vector Fields Vector (magnitude, direction) at every point in
space Example: Velocity vector field - jet stream
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Vector Fields Explained
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Examples of Vector Fields
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Scalar and vector quantities Scalar quantity is defined as a
quantity or parameter that has magnitude only. It independent of
direction. Examples: time, temperature, volume, density, mass and
energy. Vector quantity is defined as a quantity or parameter that
has both magnitude and direction. Examples: velocity, electric
fields and magnetic fields.
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A vector is represent how the vector is oriented relative to
some reference axis. Graphically, a vector is represented by an
arrow, defining the direction, and the length of the arrow defines
the vectors magnitude as shown in figure. Vectors will be indicated
by italic type with arrow on the character such as . Scalars
normally are printed in italic type such as A.
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A unit vector has a magnitude of unity (=1). The unit vector in
the direction of vector is determined by dividing A.
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By use of the unit vectors, , along x, y and z axis of a
Cartesian system, a vector quantity can be written as:
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The magnitude is defined by The unit vector is defined by
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Example 1 A vector is given as 2 + 3 sketch and determines its
magnitude and unit vector.
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Solution The magnitude of vector = 2 + 3 is
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Position and Distance Vectors A position vector is the vector
from the origin of the coordinate system O (0, 0, 0) to the point P
(x, y, z). It is shown as the vector The position vectors can be
written as:
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A distance vector is defined as displacement of a vector from
some initial point to a final point. The distance vector from P 1
(x 1, y 1, z 1 ) to P2 (x 2, y 2, z 2 ) is The distance between two
vectors is:
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Example
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Solution
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Basic Laws of Vector Algebra Any number of vector quantities of
the same type (i.e. same units) can be cmbined by basic vector
operations. For instance, two vectors are given for vector
operation below
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Vector Addition and Subtraction Two vectors may be summed
graphically by applying parallelogram rules or head-to-tail rule.
Parallelogram rule draw both vectors from a common origin and
complete the parallelogram however head-to-tail rule is obtained by
placing vector at the end of vector to complete the triangle;
either method is easily extended to three or more vectors.
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Figure show addition of two vectors follow the rules, the sum
of the addition is
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The rule for the subtraction of vectors follows easily from
that for addition, may be expressed The sign, or direction of the
second vector is reversed and this vector is then added to the
first by the rule for vector addition
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Example Solution
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Vector Multiplication Simple product Simple product multiply
vectors by scalars. The magnitude of the vector changes, but its
direction does not when the scalar is positive. It reverses
direction when multiplied by a negative scalar.
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Dot Product Dot product also knows as scalar product. It is
defined as the product of the magnitude of , the magnitude of, and
the cosine of the smaller angle, AB between and. If both vectors
have common origin, the sign of product is positive for the angle
of 0 AB 90. If the vector continued from tail of the vector, it
produce negative product since the angle is 90 AB 180.
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Commutative law : Distribution law : Associative law :
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Consider two vector whose rectangular components are given such
as Therefore yield sum of nine scalar terms, each involving the dot
product of two unit vectors. Since the angle between two different
unit vectors of the rectangular coordinate system is 90 (cos AB =0)
so
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The remaining three terms is unity because unit vector dotted
with itself since the included angle is zero (cos AB =1) Finally,
the expression with no angle is produced
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Unit vector relationships It is frequently useful to resolve
vectors into components along the axial directions in terms of the
unit vectors i, j, and k.
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The Cross Product
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Right Hand Rule 37
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Example
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Solution
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Example
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Solution
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Scalar and Vector Triple Product
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Scalar triple product The magnitude of is the volume of the
parallelepiped with edges parallel to A, B, and C. A B C ABAB
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Vector triple product The vector is perpendicular to the plane
of A and B. When the further vector product with C is taken, the
resulting vector must be perpendicular to and hence in the plane of
A and B : A B C ABAB where m and n are scalar constants to be
determined. Since this equation is valid for any vectors A, B, and
C Let A = i, B = C = j:
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x z y VECTOR REPRESENTATION: UNIT VECTORS Unit Vector
Representation for Rectangular Coordinate System The Unit Vectors
imply : Points in the direction of increasing x Points in the
direction of increasing y Points in the direction of increasing z
Rectangular Coordinate System
