Vectors

© Department of Physics, University of Guelph


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In this tutorial we will examine some of the elementary ideas concerning vectors. The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction. And it is obviously more convenient to represent both quantities by just one symbol. That is the vector.


Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. This is shown in Panel 1. . If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ. 
Panel 1

This is often simplified to just . The line and arrow above the Q are there to indicate that the symbol represents a vector. Another notation is boldface type as: Q.

Note, that since a direction is implied, . Even though their lengths are identical, their directions are exactly opposite, in fact OQ = -QO.

The magnitude of a vector is denoted by absolute value signs around the vector symbol: magnitude of Q = |Q|.

The operation of addition, subtraction and multiplication of ordinary algebra can be extended to vectors with some new definitions and a few new rules. There are two fundamental definitions.
#1 Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points, as shown in 
Panel 2.

Panel 2
#2 A vector having the same magnitude as A but in the opposite direction to A is denoted by -A , as shown in Panel 3.
Panel 3


We can now define vector addition. The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B , as illustrated in Panel 4. This is sometines referred to as the "Tip-to-Tail" method.
Panel 4

The operation of vector addition as described here can be written as C = A + B

This would be a good place to try this simulation on the graphical addition of vectors. Use the "BACK" buttion to return to this point.



Vector subtraction is defined in the following way. The difference of two vectors, A - B , is a vector C that is, C = A - B
or C = A + (-B).Thus vector subtraction can be represented as a vector addition.
The graphical representation is shown in Panel 5. Inspection of the graphical representation shows that we place the initial point of the vector -B on the final point the vector A , and then draw a line from the initial point of A to the final point of -B to give the difference C.
Panel 5


Any quantity which has a magnitude but no direction associated with it is called a "scalar". For example, speed, mass and temperature.

Many of the laws of ordinary algebra hold also for vector algebra. These laws are: Vectors can be related to the basic coordinate systems which we use by the introduction of what we call "unit vectors."
 Let us consider the two-dimensional (or x, y)Cartesian Coordinate System, as shown in
Panel 7.

Panel 7
We can define a unit vector in the x-direction by  or it is sometimes denoted by . Similarly in the y-direction we use  or sometimes . Any two-dimensional vector can now be represented by employing multiples of the unit vectors,  and , as illustrated in Panel 8.
Panel 8

The vector A can be represented algebraically by A = Ax + Ay. Where Ax and Ay are vectors in the x and y directions. If Ax and Ay are the magnitudes of Ax and Ay, then  Ax and Ay are the vector components of A in the x and y directions respectively.
The actual operation implied by this is shown in Panel 9.
Remember  (or ) and  (or ) have a magnitude of 1 so they do not alter the length of the vector, they only give it its direction.

Panel 9



The breaking up of a vector into it's component parts is known as resolving a vector. Notice that the representation of A by it's components,  Ax and Ay is not unique. Depending on the orientation of the coordinate system with respect to the vector in question, it is possible to have more than one set of components.

It is perhaps easier to understand this by having a look at an example.
Consider an object of mass, M, placed on a smooth inclined plane, as shown in Panel 10. The gravitational force acting on the object is
F = mg where g is the acceleration due to gravity. 

Panel 10

In the unprimed coordinate system, the vector F can be written as F = -Fy, but in the primed coordinate system F = -Fx'+ Fy'. Which representation to use will depend on the particular problem that you are faced with.

For example, if you wish to determine the acceleration of the block down the plane, then you will need the component of the force which acts down the plane. That is,  -Fx'which would be equal to the mass times the acceleration.



The breaking up of a vector into it's components, makes the determination of the length of the vector quite simple and straight forward.

The resolution of a vector into it's components can be used in the addition and subtraction of vectors.

Until now, we have discussed vectors in terms of a Cartesian, that is, an x-y coordinate system. Any of the vectors used in this frame of reference were directed along, or referred to, the coordinate axes. However there is another coordinate system which is very often encountered and that is the Polar Coordinate System.

The multiplication of two vectors, is not uniquely defined, in the sense that there is a question as to whether the product will be a vector or not. For this reason there are two types of vector multiplication.
The scalar product of two vectors, A and B denoted by A·B, is defined as the product of the magnitudes of the vectors times the cosine of the angle between them, as illustrated in Panel 16.
Panel 16
 Let us do an example. Consider two vectors,  and . Now what is the angle between these two vectors? This concludes our survey of the elementary properties of vectors, we have concentrated on fundamentals and have restricted ourselves to the discussion of vectors in just two dimensions. Nevertheless, a sound grasp of the ideas presented in this tutorial are absolutely essential for further progress in vector analysis.

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