MATHEMATICS TODAY: VECTOR METHODS

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 A_x, A_y, and A_z, 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 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:

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,