Introduction to vectors

1 Introduction to vectorsmc-TY-introvector-2009-1A vector is a quantity that has both a magnitude (or size) and adirection. Both of theseproperties must be given in order to specify a vector completely. In this unit we describe how towrite down vectors , how to add and subtract them, and how to use them in order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: distinguish between a vector and a scalar; understand how to add and subtract vectors ; know when one vector is a multiple of another; use vectors to solve simple problems in vector notation for two two a vector to of unit vectors in mathcentre 20091. IntroductionVector quantities are extremely useful in physics. The important characteristic of a vector quan-tity is that it has both a magnitude (or size) and a of these properties must begiven in order to specify a vector example of a vector quantity is a displacement.

2 This tell us how far away we are from a fixedpoint, and it also tells us our direction relative to that example of a vector quantity is velocity. This is speed, in a particular direction. Anexample of velocity might be 60 mph due quantity with magnitude alone, but no direction, is not a vector . It is called example of a scalar is distance. This tells us how far we are from a fixed point, but does notgive us any information about the direction. Another example of a scalar quantity is the mass ofan PointA vector has both magnitude and direction, and both these properties must be given in orderto specify it. A quantity with magnitude but no direction is called a Representing vector quantitiesWe can represent a vector by a line segment. This diagram shows two have used a small arrow to indicate that the first vector is pointing fromAtoB. A vectorpointing fromBtoAwould be going in the opposite we represent a vector with a small letter such asa, in a bold typeface.

3 This iscommon in textbooks, but it is inconvenient in writing, we normally put a barunderneath, or sometimes on top of, the letter:aora. In speech, we call this the vector a-bar . mathcentre 20093. Position vectorsSometimes vectors are referred to a fixed point, an origin. Such a vector is called a positionvector. So we might refer to the position vector of a pointPwith respect to an originO. Inwriting, might putOPfor this vector . Alternatively, we could write it asr. These two expressionsrefer to the same Some notation for vectorsWhat does it mean if, for two vectors ,a=b? This means first that the length ofaequals thelength ofb, so that the two vectors have the same magnitude. But it also means thataandbare in the same direction. How can we write this down more succinctly?If two vectors are in the same direction , then they are parallel. We write this down length, if we have a vectorAB, we can write its length asABwithout the bar.

4 Alternatively,we can write it as|AB|. The two vertical lines give us the modulus, or size of, the vector . If wehave a vector written asa, we can write its length as either|a|with two vertical lines, or asain ordinary type (or without the bar). This is why it is very important to keep to the conventionthat has been adopted in order to distinguish between a vector and its PointThe length of a vectorABis written asABor|AB|,and the length of a vectorais written asa(in ordinary type, or without the bar) or as|a|.If two vectorsaandbare parallel, we mathcentre 20095. adding two vectorsOne of the things we can do with vectors is to add them together. We shall start by addingtwo vectors together. Once we have done that, we can add any number of vectors together byadding the first two, then adding the result to the third, and so order to add two vectors , we think of them as displacements. We carry out the first dis-placement, and then the second.

5 So the second displacement must start where the first + babThe sum of the vectors ,a+b(or theresultant, as it is sometimes called) is what we get whenwe join up the triangle. This is called thetriangle lawfor adding is another way of adding two vectors . Instead of makingthe second vector start wherethe first one finishes, we make them both start at the same place, and complete a is called theparallelogram lawfor adding vectors . It gives the same result as the trianglelaw, because one of the properties of a parallelogram is thatopposite sides are equal and in thesame direction, so thatbis repeated at the top of the + bKey PointWe can add two vectorsaandbby makingbstart whereafinishes, and completing thetriangle. Alternatively, we can makeaandbstart at the same place, and take the diagonal ofthe mathcentre 20096. Subtracting two vectorsWhat isa b? We think of this asa+ ( b), and then we ask what bmight mean. This willbe a vector equal in magnitude tob, but in the reverse bNow we can subtract two vectors .

6 Subtractingbfromawill be the same asadding ba bKey Pointa bmeansa+ ( b)7. adding a vector to itselfWhat happens when you add a vector to itself, perhaps severaltimes? We write, for example,a+a+a= the same way, we would writena=a+..+a mathcentre 2009 Key PointA vectornais in the same direction as the vectora, butntimes as vectors of unit lengthThere is one more piece of notation we shall use when writing vectors . Ifais any vector , we shallwrite ato represent a unit vector in the direction ofa. A unt vector is a vector whose length is1, so that| a|= notation gives us another way of writing the vectora: we can write it asa a, so that it isthe lengthamultiplied by the unit vector PointA unit vector in the direction of the vectorais written as a, so thata=a Using vectors in geometryExampleThere is a useful theorem in geometry called themid-point theorem. In this theorem, we taketwo pointsAandB, defined with respect to an originO.

7 Let us writeafor the position vectorofA, andbfor the position vector ofB. We can joinAandBwith a line, to give a take the mid-pointMof the lineOA, and the mid-pointNof the lineOB, and joinMtoNwith a line. Can we say anything about the relationship between the lineMNand the lineAB? mathcentre 2009 ABOMNabWe can answer this very easily with vectors . We can write the vector for the line segmentABasAO+OB. NowAOis the reverse of the vectora, so it is a. AndOBis the same as thevectorb. ThereforeAB=AO+OB= ( a) +b=b aboutMN? Following the same reasoning, this isMO+ON. But what isMO?This is avector half the length ofAO, and in the same direction, so it must be12( a). In the same way,ONis in the same direction asOB, but is half the length, so it must be12b. ThereforeMN=MO+ON=12( a) +12b=12(b a).Now we can compareABandMN. From our calculation, we can see thatMNis12AB. So,as this is a vector equation, it tells us two things. First, ittells us about magnitude, so thatMN=12AB.

8 Also, it tells us thatMNandABmust be in the same direction, so is called the mid-point theorem for a triangle. It states that if you join the mid-points oftwo sides of a triangle then the resulting line is equal to half of the third side of the triangle, andis parallel to can apply the mid-point theorem to a quadrilateral, or indeed to any four points in space, togive an interesting geometrical result. We shall call the four pointsA,B,CandD. We shallalso give labels to the mid-points of the four sides: we shallcall the mid-pointsP,Q, let us join the four mid-points, to make a new shapeP QRS. What kind of shape is this? mathcentre 2009We can identify the shape by joining the we apply the mid-point theorem to triangleABC, we see thatP Q= if we apply the mid-point theorem to the triangleADC, we also see thatRS= we combine these two equations, we then obtainP Q= this is a vector equation, and so it tells us two things. First, it tells us that the length ofP Qis the same as the length ofRS.

9 And secondly, it tells us that the direction ofP Qis thesame as the direction ofRS, so thatP QandRSare parallel. But having two parallel sides ofequal length is a property which defines a parallelogram, andso the shapeP QRSmust be shall now use vectors to prove one more two pointsAandB, having position vectorsa,bwith respect to an originO. Draw thelineAB, and take a pointPon that line which divides it in the ratio ofmton. What is theposition vector ofPwith respect toO?.ABOP abrmnWe can use the same method that we used before. We know thatOP=OA+AP ,(1)and we also know thatOA=a. But what isAP?NowAPis in the same direction asAB, and their lengths are in the ratio ofmtom+n. SoAP=mm+nAB.(2)We also know thatAB=AO+OB=b mathcentre 2009 Now we can put these three statements together, replacingAPin equation (1) by using equa-tion (2), and replacingABin equation (2) by using the equation (3), so that everythingwill bewritten in terms ofaandb.

10 This gives usOP=a+mm+n(b a).Putting all this over a common denominator then givesOP=(m+n)a+m(b a)m+ we expand the brackets, the termmawill cancel with the termm( a), and so finally we haveOP=na+mbm+ formula gives us a way of calculating the position vector of the pointP. For instance, ifmandnwere both1thenPwould be the mid-point ofAB. The position vector of the midpointwould be(a+b)/2. As another example, ifm= 2andn= 1, so thatPwas two-thirds of theway along the line, then the position vector ofPwould be(a+ 2b) The vectorais shown the vectors2a,3a,12aand In OAB,OA=aandOB=b. In terms ofaandb,(a) What isAB?(b) What isBA?(c) What isOP, wherePis the midpoint ofAB?(d) What isAP?(e) What isBP?(f) What isOQ, whereQdividesABin the ratio 2:3?3. What is meant by a unit vector ?4. Ifeis a unit vector , what is the length of3e?5. In ABC,AB=a,BC=b,CA=c. What isa+b+c? mathcentre 2a122.(a)b a(b)a b(c)12(a+b)(d)12(b a)(e)12(a b)(f)35a+25b3.