The concept of vectors is introduced now because we can no longer deal with simple values. A vector value is a term that has both a magnitude and direction. In terms of a velocity it is a speed with a magnitude of distance per unit time in some specific direction. One can feel this concept when spinning a bicycle wheel and attempting to twist it vector. Mr. Kolar in my high school taught this to me. Even to a baseball player it's how to chuck a mean curve. The problem: what are the results when you multiply two directions?
Vector notation is generally performed by darker printing. For example; the notation for a specific value is V, the vector notation for a value is V.
Also, vectors can be noted by a magnitude and a vector subscript such as:
Where V is the value, and 
is the subscripted vector direction component.[NOTE: The samples are pictures. Such proper scientific notation may be missing because of the way HTML programers wrote HTML browsers. In the world of science, one quickly sees the limitations of HTML. Most the rest of the writing, at least for a while, will have subscripts just along side the notation: ie, Vi & ii. Good Luck!]
Real world vector notation would be given by three different direction and a magnitude in each of the three directions. For example;
Where 
is the magnitude in the direction x and is the unit vector showing direction. (Note: V here refers to vector and not voltage.)
The magnitude is:
and the direction cosines satisfy the proportion;
Vector addition is the same as standard addition. Simply the magnitudes are added together and the direction components are the same.
Vectors follow three different laws. These three laws are called associative, commutative and distributive laws. Give u, v, and w are all vectors and c, d, 1, and 0 are all real numbers, then;
Associative laws:
Commutative law:
Distributive laws:
Multiplication of vector quantities is different than standard multiplication. There are two types of multiplication systems which control vectors. They are dot products and cross products.
Dot products. The term is referred to as scalar product.
Alpha is the angle between the two vectors. Where:
Scalar multiplication is commutative.
and the unit vectors are:
Vector cross products is the second type of vector multiplication. This type of multiplication has very different outcomes than standard multiplication. The term 
X 
is stated as either cross or carrying intor and is given by:
Theta is the angle from to and l is a unit vector perpendicular to the plane of and and so directed that it follows the right hand rule driven in the direction of l by 90 degrees.
[Just so you see what HTML does to some math notation, this error will look like V2 = a2i + b2j + c2k.]
Vector operator.
The vector operator is a term designed to give three dimensional vector differentiation.
As an example; the vector operator on M in a courtesan system:
When a spherical system is considered, vector formulas become much more complex. Instead of a system where the distances are given by x, y, and z, the system is defined by r for the radius and two angles "phi" and "theta"
Typically in spherical coordinates the cartesian unit vectors i, j, & k are r, phi, & theta. The unit vector relationships to each other are the same in spherical coordinate as they are in cartesian coordinates.
The following relationships apply:
As an example; the vector operator on M in a spherical system:
The vector operator can be used more than once in a matrix solution. Curl and grad functions factor down well. Vectors can be differentiated and integrated.
A simple rule of hand; 90 degree vector multiplication; the right hand rule. The thumb is in one direction of one vector, the first finger is pointing 90 degrees, and the middle finger pointing 90 degrees in the third direction. Out is the product of the angels of one and two. The little finger, if it is curled as mine, is the direction of the curl.
For those who use the right hand rule to remember how current, magnetic field, and force go, and possibly some may remember the left hand rule for generators, I now give you the two hand rule.
A sphere is the single only mathematical phenomena which allows for a complete vector transformation.
In order to maintain equilibrium in a spherical system we introduce a number which requires imagination. It is some times referred to as an imaginary number. The number has real affects. The number is the square root of minus one. It seems, the imaginary number squared has some real solutions.
The two hand rule is simply the representation of the square root of minus one for the left hand. Put up your right hand and extend the fingers to the right hand rule. Now, make the left hand fingers sort of look like the right hand fingers. Put your hands back to back so one thumb is pointing up and the other thumb is pointing down. The total spherical representation of expanding vectors is represented by both hands.
Many can not visualize a vector force. Some real vectored forces are water, circling, while it is going down in the sink; a top spinning while leaning to the side and oscillating; a Frisbee flying; and riding a bicycle, are all vector force reactions.
In the real world, there are two considerations which perplex the individual. They are: how big is big and how small is small? When considering flux, the two questions arise as how large does a body need to be to contain all the flux and what happens to the flux as a body shrinks to zero volume?
What happens to the force when one shrinks the volume of a body to zero is a practical question which occurs in many physical investigations. So many times in fact that it was given a descriptive name, divergence. The divergence of a vector is usually referred as Div V. Divergence obeys the distributive law.
Divergence in a cartesian system:
Divergence in a spherical system:
The divergence of two vector functions:
Another mathematical situation which requires definition is created when one wants to find the vector relation ship of two points within an electric field as the points change position. The field intensity changes from one point to the other and so does the direction of field flux. Because the field is a scalar function at a point this uses a vector operator on a scalar point function. Thus, the possibility for total confusion because the vector operator operating on a vector is different from a vector operator operating on a scalar but it prints and looks really close to the same equations. Basically, V is not the same as V. V being a vector and V being a scalar. Gradient of V is called, grad V.
Gradient in a cartesian system:
In the above equation S was used instead of V or M. As one can see, if the same letter were used as the vector operator operating on a vector, it may be difficult to see what the difference is.
Gradient in a spherical system:
Another term used a lot in magnetic is the curl. Because this relationship is very common in magnetic and because other letters have been used to describe vectors in general, H will be used here. H is the magnetic field intensity given in amperes per meter. However, any vector will fit. Simply put, curl is the cross product of the vector operator and a vector.
It should be noted, the curl was derived by a current flowing in a conductor and the net current flowing out of the conductor is zero. This is because in a wire conductor, the net current flows in a closed circuit. Thus, current is flowing in a closed path and none of the current is lost. Another reason for the current flowing in a conductor as the model for curl is that experiments can verify the theoretical results because current through wires and the associated magnetic can be easily produced and measured.
Curl in a cartesian system:
Curl in a spherical system:
Curl may also be in the form of a determinant:
Laplacian: The second order differential of the operator operating on a vector.
Laplacian in a cartesian system:
Laplacian in a spherical system:
Some vector operation identities: Given: vectors A, B, & C & none vectors U & W.
Some examples of the way these identities are stated:
Where div is divergence, grad is gradient, and curl is curl.
The major functions here are the vector operator operating on a vector, the vector operator operating on a scalar, the dot vector operation on a vector, the cross vector operation on a vector and the second order differential of the vector operator operating on a scaler. Note: Again, I stress the use of "X" means cross product here. HTML doesn't have any way of handling cross products either. Watch out! And, think three dimensional vectors all the time while reading this document.
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