Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 25
Vektor
A vector is formally defined as an element of a vector space. In the
commonly encountered vector space
(i.e., Euclidean n-space), a
vector is given by
coordinates and can be specified
as
. Vectors are sometimes referred to by the
number of coordinates they have, so a 2-dimensional vector
is
often called a two-vector, an -dimensional vector is often called an nvector, and so on.
Vectors can be added together (vector addition), subtracted (vector
subtraction) and multiplied by scalars (scalar multiplication). Vector
multiplication is not uniquely defined, but a number of different types of
products, such as the dot product, cross product, and tensor direct
product can be defined for pairs of vectors.
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A vector from a point
to a point
is denoted
, and a
vector
may be denoted , or more commonly, . The point
is
often called the "tail" of the vector, and
is called the vector's
"head." A vector with unit length is called a unit vector and is denoted
using a hat, v^ .
When written out componentwise, the notation
generally refers
to
. On the other hand, when written with a
subscript, the notation
(or
) generally refers
to
.
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An arbitrary vector may be converted to a unit vector by dividing
by its norm (i.e., length),
giving
(
2
)
A zero vector, denoted , is a vector of length 0, and thus has all
components equal to zero.
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(
1
)
Vector Addition
Vector addition is the operation of adding two or
more vectors together into a vector sum.
The so-called parallelogram law gives the rule for vector addition
of two or more vectors. For two vectors
and
, the vector
sum
is obtained by placing them head to tail and drawing
the vector from the free tail to the free head. In Cartesian
coordinates, vector addition can be performed simply by adding
the corresponding components of the vectors, so
if
and
,
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Dot Product
The dot product can be defined for two
vectors and by
where
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is the angle between the vectors and
is the norm.
Cross Product
For vectors
the cross product is defined by
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and
in
,
where
is a right-handed, i.e., positively oriented,
orthonormal basis. This can be written in a shorthand notation that
takes the form of a determinant
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• Kerjakan latihan dalam modul soal
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