This tutorial was arisen on the communication of the one I found in the room of Physics, university of Guelph.

In this indict we will examine several of the elementary concepts concerning vectors. The reason for this introduction to vectors is that many ideas in science, because that example, displacement, velocity, acceleration, force have a size or magnitude, but also they have associated with lock the idea that a direction. And also it is obviously an ext convenient to denote both amounts by just one symbol. That is the vector.

We want you to save in mind, the vectors room measurable physical quantaties. Because of this they carry out not rely on united state — observers. In particular, they perform not rely on our an option of the name: coordinates system. However, the means we stand for a vector in any kind of given device of coordinates does count on the name: coordinates system.

### Vectors

Figure 1

Graphically, a vector is representedby one arrow, specifying the direction, and the size of the arrowhead definesthe vector"s magnitude. This is shown in Fig. 1. If we denote one endof the arrowhead by the beginning O and the tip of the arrowhead by Q. Climate the vectormay be stood for algebraically by OQ.

This is often simplified to just \$vecQ\$ or \$arQ\$The arrow and also the line over the Q space there to suggest that the symbol representsa vector. One more notation is boldface kind as: \$f Q\$

because a direction is implied, \$f OQ ot=f QO\$Even though their lengths are identical, your directions are exactly opposite,in reality \$f OQ=-f QO\$. The magnitude of a vector is denoted by pure value signs aroundthe vector symbol: size of \$f Q\$ is created as \$|f Q|\$ (or size of \$f OQ\$ is composed as \$|f OQ|\$)
There room two fundamental definitions.

You are watching: The value of the dot product of two vectors depends on the particular coordinate system being used.

Figure 2

2 vectors, \$vecA\$ and \$vecB\$ room equal if they have actually the very same magnitude and direction, nevertheless of whether they have the very same initial points, as presented in figure 2.

Figure 3

A vector having actually the very same magnitude as \$vecA\$ however the the opposite direction is denoted by \$-vecA\$, as shown in number 3.

The operation of addition, subtraction and multiplication of ordinary algebra can be prolonged to vectors with some new definitions and also a couple of new rules.

Figure 4

We deserve to now define vector addition. The sum of 2 vectors, \$vecA\$ and also \$vecB\$, is a vector \$vecC\$, i m sorry is obtained by place the initial allude of \$vecB\$ ~ above the final allude of \$vecA\$, and then illustration a line from the initial point of \$vecA\$ come the final allude of \$vecB\$, as portrayed in figure 4. This is sometimes referred to together the "Tip-to-Tail" method.

The operation of vector addition as described here is composed as \$vecC=vecA+vecB\$.

Vector subtraction is defined in the following way. The difference of 2 vectors, \$vecA-vecB\$ , is a vector \$vecC\$ the is, \$vecC=vecA-vecB\$ or \$vecC=vecA+(-vecB)\$. Hence vector subtraction have the right to be stood for as a vector addition.

Figure 5
The graphical depiction is presented in number 5. Inspection of the graphical representation reflects that we location the initial suggest of the vector \$-vecB\$ on the final point the vector \$vecA\$, and then attract a heat from the initial point of \$vecA\$ to the final point of \$-vecB\$ to offer the distinction \$vecC\$.
This would certainly be a an excellent place to shot this simulation on the graphical addition of vectors. Usage the "BACK" buttion to go back to this point.

### Scalars

any quantity which has a magnitude but no direction connected with the is called a "scalar". For example, speed, mass and temperature. The product that a scalar, \$m\$ say, time a vector \$vecA\$, is one more vector, \$vecB\$, wherein \$vecB\$ has the very same direction as \$vecA\$ however the size is changed, the is, \$|vecB=m|vecA|\$.

### Vector Algebra

numerous of the regulations of ordinary algebra hold likewise for vector algebra. These laws are: Commutative law for Addition: \$vecA+vecB=vecB+vecA\$. Associative law for Addition: \$vecA+(vecB+vecC)=(vecA+vecB)+vecC\$. Commutative law for Multiplication: \$mvecA=vecAm\$, where \$m\$ is a scalar. Associative law for Multiplication: \$(m+n)vecA=mvecA+nvecA\$, wherein \$m\$ and \$n\$ room two various scalars. Distributive Law: \$m(vecA+vecB)=mvecA+mvecB\$.
evidence of the associative regulation
Associative legislation is shown in figure 6. If we include \$vecA\$ and also \$vecB\$ and also we acquire a vector \$vecE\$. And likewise if \$vecB\$ is included to \$vecC\$, we acquire \$vecF\$. Currently \$vecD=vecE+vecC=vecA+vecF\$. Replacing \$vecE\$ with \$(vecA+vecB)\$ and also \$vecF\$ v \$(vecB+vecC)\$, we gain \$vecA+(vecB+vecC)=(vecA+vecB)+vecC\$ The legislation is verified.
Figure 6
stop now and make sure that you monitor the above proof. Prove the rest of the laws by yourself.
these laws permit the manipulation the vector quantities in much the same way as plain algebraic equations.

### Coordinates

Vectors deserve to be regarded the basic coordinate solution which we use by the introduction of what we call "unit vectors." A unit vector is one which has actually a magnitude of \$1\$ and also is often indicated by placing a cap (or circumflex) on top of the vector symbol, for example \$hata\$, \$|hata|=1\$. I emphasize the \$hata\$ is a vector, it has direction. Thequantity \$hata\$ is review as "a hat" or "a unit".
Figure 7
allow us determined a two-dimensional (or \$x\$,\$y\$) Cartesian coordinate System, as presented in figure 7.

Important point: Vector represents a physical quantitywhich exist independent the us. Therefore vector chin does not rely on ours choice that the coordinate system. The works with of the vector, however, execute depend ~ above the mechanism of coordinates.

Figure 8

We can specify a unit vector in the \$x\$ direction \$hatx\$,it is also denoted by \$hati\$. Likewise in the \$y\$-directionwe usage \$haty\$or\$hatj\$.Any two-dimensional vector can now be stood for by employing multiplesof the unit vectors,\$hatx\$ and\$haty\$,as illustrated in number 8.

The vector A have the right to be stood for algebraically through \$vecA=vecA_x + vecA_y\$. Wherein \$vecA_x\$ and also \$vecA_y\$are vectors in the \$x\$ and also \$y\$ directions. If \$A_x\$ and \$A_y\$are the magnitudes the \$vecA_x\$ and \$vecA_y\$,then \$vecA_x=A_x hatx\$ and \$vecA_y=A_y hatj\$. \$A_x\$ and \$A_y\$ room called contents of vector \$vecA\$ in \$x\$ and also \$y\$ direction respectively. Vectors \$vecA_x\$ and also \$vecA_y\$ space sometimes dubbed vector components of the vector \$vecA\$.

The vector \$vecA\$ in the chosen mechanism of works with then deserve to be represented as \$vecA=A_x hatx+A_y hatj\$.

Figure 9

The actual operation implied through this is shownin figure 9.

Remember:\$hatx\$ (or \$hati\$) and also \$haty\$ (or \$hatj\$) have actually a magnitude of \$1\$, so they only provide a vector that is direction.

### Resolving a vector

The break up that a vector into it"s component parts is well-known as addressing a vector. An alert that the depiction of \$vecA\$ by it"s components, \$A_x hati\$and \$A_y hatj\$is not unique. Depending on the orientation the the coordinate mechanism withrespect come the vector in question, it is possible to have an ext than oneset of components.

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It is perhaps much easier to understand this by having actually a look at an example.

Figure 10

Consider an object of mass, \$M\$, put on a smoothinclined plane, as shown in figure 10. The gravitational pressure acting onthe thing is a vector \$vecF=mvecg\$,where \$vecg\$ is the acceleration of totally free fall.

In the unprimed name: coordinates system, the vector \$vecF\$ have the right to be written as \$vecF=-F_y hatj\$,but in the primed coordinate mechanism the same vector \$vecF=-F_x" veci"+F_y"hatj"\$.

Again the vector \$vecF\$ is the exact same vector in both situations (after all it is a actual force!), yet our representation of that changed, as we changed the name: coordinates system.Which representation to use will rely on the specific problem thatyou are confronted with.

For example, if you great to recognize the acceleration of the blockdown the plane, climate you will need the component of the force which actsdown the plane. The is, \$-F_x"hati"\$, whichwould be same to the mass time the acceleration.

### Components, magnitude and also angle

Figure 11

The break up the a vector right into it"s components, provides the determination of the length of the vector quite simple and directly forward.

due to the fact that \$vecA=A_x hati+A_y hatj\$, then making use of Pythagorus" organize we discover the magnitude of the vector \$vecA\$: \$|vecA|=sqrtA_x^2+A_y^2\$.

for example, if \$vecA=3hati+4hatj\$, climate \$|vecA|=sqrt3^2+4^2=sqrt25=5 \$.

we can likewise find the angle in between the vector \$vecA\$ and also say \$x\$ direction. From trigonometry we have \$cos (Theta)=A_x/|vecA|\$.

very often in vector problems you will know the length, the is, the magnitude of the vector and you will additionally know the direction of the vector. From these you will have to calculate the Cartesian components, that is, the \$x\$ and \$y\$ components.

The situation is illustrated in figure 11. Let united state assume that the magnitude of \$vecA\$ and the angle \$Theta\$ are given; what we wish to recognize is, what are \$A_x\$ and \$A_y\$?

Again the very same trigonometry gives: \$cos (Theta)=A_x/|vecA|\$ as such \$A_x=|vecA|cos (Theta)\$, and similarly \$A_y=|vecA|cos (90^circ-Theta)=|vecA|sin (Theta)\$,

The resolution that a vector into it"s components can be used in the enhancement and individually of vectors. To highlight this allow us consider an example, what is the amount of the following three vectors?

egineqnarray&&vecA=A_x hati+A_yhatj\&&vecB=B_x hati+B_yhatj\&&vecC=C_x hati+C_yhatjendeqnarray

By solving each that these three vectors into their materials we see that the result is number 12.

egineqnarray&&D_x=A_x+B_x+C_x\&&D_y=A_y+B_y+C_yendeqnarray

so because that the vector \$vecD\$ we gain \$vecD=D_x hati+D_yhatj\$.

now we can calculate the magnitude of the vector \$vecD\$, as well as the angle in between the vector \$vecD\$ and, say, \$x\$ direction.