**
VECTOR METHODS **

**Areas of focus:**

- Vectors and vector addition
- Unit vectors
- Base vectors and vector components
- Rectangular coordinates in 2-D
- Rectangular coordinates in 3-D
- A vector connecting two points
- Dot product
- Cross product
- Triple product
- Triple vector product

** **

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.

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 **u**_{1}, **u**_{2},
and **u**_{3}, each along the direction of one of the base
vectors **e**_{1}, **e**_{2}, and **e**_{3},
so that

Each one of the vectors **u**_{1}, **u**_{2}, and
**u**_{3} is parallel to one of the base vectors and can be
written as scalar multiple of that base. Let *u*_{1}, *u*_{2},
and *u*_{3} denote these scalar multipliers such that one has

The original vector **u** can
now be written as

The scalar multipliers *u*_{1}, *u*_{2}, and *
u*_{3} are known as the components of **u** in the base
described by the base vectors **e**_{1}, **e**_{2},
and **e**_{3}. If the base vectors are unit vectors, then the
components represent the lengths, respectively, of the three vectors **u**_{1},
**u**_{2}, and **u**_{3}. 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 *A _{x},
A_{y}, *and

** **

** **

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

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 90^{o} 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 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,

** **

The cross product has the following properties.

**Rectangular coordinates:**

When working in rectangular coordinate systems,
the cross product of vectors **a** and **b** given by

** **

can be evaluated using the rule

One can also use direct multiplication of the base vectors using the relations

**
**

** **

The triple product of vectors **a**, **b**, and **c** is given
by

** **

The value of the triple product is equal to the volume of the parallelepiped constructed from the vectors. This can be seen from the figure since

The triple product has the following properties

**Rectangular coordinates:**

Consider vectors described in a rectangular coordinate system as

The triple product can be evaluated using the relation

The triple vector product has the properties