Vector Algebra: - Chia Sẻ Kiến Thức Điện Máy Việt Nam

VECTOR METHODS

Areas of focus:

Vectors and
vector addition:

A scalar is a quantity like mass or temperature that only has a magnitude.
On the other had, a vector is a mathematical object that has magnitude and
direction. A line of given length and pointing along a given direction, such as
an arrow, is the typical representation of a vector. Typical notation to
designate a vector is a boldfaced character, a character with and arrow on it,
or a character with a line under it (i.e., ). The magnitude of a vector is
its length and is normally denoted by

or A.

Addition of two vectors is accomplished by laying the vectors head to tail
in sequence to create a triangle such as is shown in the figure.

The following rules apply in vector algebra.

where P and Q are vectors and a is a scalar.

Unit vectors:

A unit vector is a vector of unit length. A unit vector is sometimes denoted
by replacing the arrow on a vector with a “^” or just adding a
“^” on a boldfaced character (i.e., ). Therefore,

Any vector can be made into a unit vector by dividing it by its length.

Any vector can be fully represented by providing its magnitude and a unit
vector along its direction.

Base vectors and
vector components:

Base vectors are a set of vectors selected as a base to represent all other
vectors. The idea is to construct each vector from the addition of vectors
along the base directions. For example, the vector in the figure can be written
as the sum of the three vectors u1, u2, and
u3, each along the direction of one of the base vectors e1,
e2, and e3, so that

Each one of the vectors u1, u2, and u3
is parallel to one of the base vectors and can be written as scalar multiple of
that base. Let u1, u2, and u3
denote these scalar multipliers such that one has

The original vector u can
now be written as

The scalar multipliers u1, u2, and u3
are known as the components of u in the base described by the base
vectors e1, e2, and e3.
If the base vectors are unit vectors, then the components represent the
lengths, respectively, of the three vectors u1, u2,
and u3. If the base vectors are unit vectors and are mutually
orthogonal, then the base is known as an orthonormal, Euclidean, or Cartesian
base.

A vector can be resolved along any two directions in a plane containing it.
The figure shows how the parallelogram rule is used to construct vectors a
and b that add up to c.

In three dimensions, a vector can be resolved along any three non-coplanar
lines. The figure shows how a vector can be resolved along the three directions
by first finding a vector in the plane of two of the directions and then
resolving this new vector along the two directions in the plane.

When vectors are represented in terms of base vectors and components,
addition of two vectors results in the addition of the components of the
vectors. Therefore, if the two vectors A and B are represented by

then,

Rectangular
components in 2-D:

The base vectors of a rectangular x-y coordinate system are given by
the unit vectors and along the x and y
directions, respectively.

Using the base vectors, one can represent any vector F as

Due to the orthogonality of the bases, one has the following relations.

Rectangular
coordinates in 3-D:

The base vectors of a rectangular coordinate system are given by a set of
three mutually orthogonal unit vectors denoted by , , and that
are along the x, y, and z coordinate directions,
respectively, as shown in the figure.

The system shown is a right-handed system since the thumb of the right hand
points in the direction of z if the fingers are such that they represent
a rotation around the z-axis from x to y. This system can
be changed into a left-handed system by reversing the direction of any one of
the coordinate lines and its associated base vector.

In a rectangular coordinate system the components of the vector are the
projections of the vector along the x, y, and z
directions. For example, in the figure the projections of vector A
along the x, y, and z directions are given by Ax, Ay,
and Az, respectively.

As a result of the Pythagorean theorem, and the orthogonality of the base
vectors, the magnitude of a vector in a rectangular coordinate system can be
calculated by

Direction cosines:

Direction cosines are defined as

where the angles , , and are the
angles shown in the figure. As shown in the figure, the direction cosines
represent the cosines of the angles made between the vector and the three
coordinate directions.

The direction cosines can be calculated from the
components of the vector and its magnitude through the relations

The three direction cosines are not independent
and must satisfy the relation

This results form the fact that

A unit vector can be constructed along a vector
using the direction cosines as its components along the x, y, and
z directions. For example, the unit-vector along the vector A
is obtained from

Therefore,

A vector
connecting two points:

The vector connecting point A to point B
is given by

A unit vector along the line A-B can be obtained from

A vector F along the line A-B and of magnitude F can
thus be obtained from the relation

Dot product:

The dot product is denoted by “” between two vectors. The
dot product of vectors A and B results in a scalar given by the
relation

where is the angle between the two vectors. Order is not important in
the dot product as can be seen by the dot products definition. As a result one
gets

The dot product has the following properties.

Since the cosine of 90o is zero, the dot product of two
orthogonal vectors will result in zero.

Since the angle between a vector and itself is zero, and the cosine of zero
is one, the magnitude of a vector can be written in terms of the dot product
using the rule

Rectangular coordinates:

When working with vectors represented in a
rectangular coordinate system by the components

then the dot product can be evaluated from the
relation

This can be verified by direct multiplication of
the vectors and noting that due to the orthogonality of the base vectors of a
rectangular system one has

Projection of a vector onto a line:

The orthogonal projection of a vector along a line
is obtained by moving one end of the vector onto the line and dropping a
perpendicular onto the line from the other end of the vector. The resulting
segment on the line is the vector’s orthogonal projection or simply its
projection.

The scalar projection of vector A along the
unit vector is the length of the orthogonal projection A
along a line parallel to , and can be evaluated using the dot product. The
relation for the projection is

The vector projection of A along the unit
vector simply multiplies the scalar projection by the unit vector to
get a vector along . This gives the relation

The cross
product:

The cross product of vectors a and b is a vector perpendicular
to both a and b and has a magnitude equal to the area of the
parallelogram generated from a and b. The direction of the cross
product is given by the right-hand rule . The cross product is denoted by a
“” between the vectors

Order is important in the cross product. If the order of operations changes
in a cross product the direction of the resulting vector is reversed. That is,