The vector product - mathcentre.ac.uk

1 The vector productmc-TY-vectorprod-2009-1 One of the ways in which two vectors can be combined is known asthevector product . Whenwe calculate the vector product of two vectors the result, asthe name suggests, is a this unit you will learn how to calculate the vector product and meet some geometrical 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: define the vector product of two vectors calculate the vector product when the two vectors are given in cartesian form use the vector product in some geometrical of the vector properties of the vector vector product of two vectors given in cartesian applications of the vector mathcentre 20091. IntroductionOne of the ways in which two vectors can be combined is known asthevector product . Whenwe calculate the vector product of two vectors the result, asthe name suggests, is a this unit you will learn how to calculate the vector product and meet some geometrical Definition of the vector productStudy the two vectorsaandbdrawn in Figure 1.

2 Note that we have drawn the two vectors sothat their tails are at the same point. The angle between the two vectors has been labelled .ab Figure 1. Two vectorsaandbdrawn so that the angle between them is .As we stated before, when we find avector productthe result is a vector . We define themodulus, or magnitude, of this vector as|a||b|sin so at this stage, a very similar definition to the scalar product , except now the sine of appearsin the formula. However, this quantity is not a vector . To obtain a vector we need to specify adirection. By definition the direction of the vector productis such that it is at right angles tobothaandb. This means it is at right angles to the plane in whichaandblie. Figure 2 showsthat we have two choices for such a 2. There are two directions which are perpendicular to convention is that we choose the direction specified by the right hand screw rule. Thismeans that we imagine a screwdriver in the right hand. The direction of the vector product mathcentre 2009the direction in which a screw would advance as the screwdriver handle is turned in the sensefromatob.

3 This is shown in Figure na b Figure 3. The direction of the vector product is determined by the right hand screw let a unit vector in this direction be labelled n. We then define the vector product ofaandbas follows:Key PointThevector productofaandbis defined to bea b=|a||b|sin nwhere|a|is the modulus, or magnitude ofa,|b|is the modulus ofb, is the angle betweenaandb, and nis a unit vector , perpendicular to bothaandbin a sense defined by the right hand screw people find it helpful to obtain the direction of the vector product using the right handthumb rule. This is achieved by curling the fingers of the right hand in the direction in whichawould be rotated to meetb. The thumb then points in the direction ofa another view is to align the first finger of the right hand witha, and the middle finger withb. If these two fingers and the thumb are then positiioned at right-angles, the thumb points inthe direction ofa b. Try this for that the symbol for the vector product is the times sign,or cross , and so we sometimesrefer to the vector product as the cross product .

4 Either namewill do. Some textbooks and someteachers and lecturers use the alternative wedge symbol . mathcentre 20093. Some properties of the vector productSuppose, for the two vectorsaandbwe calculate the product in a different order. That is,suppose we want to findb a. Using the definition ofb aand using the right-hand screw ruleto obtain the required direction we findb a=|b||a|sin ( n)We see that the direction ofb ais opposite to that ofa bas shown in Figure 4. Sob a= a bSo the vector product isnot commutative. In practice, this means that the order in which wedo the calculation ais in the opposite direction toa ab n-na b ab Figure 4. The direction ofb ais opposite to that ofa PointThe vector product isnot a= a bAnother property of the vector product is that it isdistributive over addition. This means thata (b+c) =a b+a cAlthough we shall not prove this result here we shall use it later on when we develop an alternativeformula for finding the vector mathcentre 2009 Key PointThe vector product isdistributive over addition.

5 This meansa (b+c) =a b+a cEquivalently,(b+c) a=b a+c aThe vector product of two parallel vectorsExampleSuppose the two vectorsaandbare parallel. Strictly speaking the definition of the vectorproduct does not apply, because two parallel vectors do not define a plane, and so it does notmake sense to talk about a unit vector nperpendicular to the plane. But if we nevertheless writedown the formula, we can see what the answer ought to be:a b=|a||b|sin n=|a||b|sin 0 n=0becausesin 0 = 0. So, when two vectors are parallel wedefinetheir vector product to be thezero vector , PointFor two parallel vectorsa b=04. The vector product of two vectors given in cartesian formWe now consider how to find the vector product of two vectors when these vectors are given incartesian form, for example asa= 3i 2j+ 7kandb= 5i+ 4j 3kwherei,jandkare unit vectors in the directions of thex,yandzaxes of all we need to develop a few results in the following mathcentre 2009 ExampleSuppose we want to findi j. The vectorsiandjare shown in Figure 5.

6 Note that becausethese vectors lie along thexandyaxes they must be 5. The unit vectorsi,jandk. Note thatkis a unit vector perpendicular angle betweeniandjis90 , andsin 90 = 1. Further, if we apply the right hand screw rule,a vector perpendicular to bothiandjisk. Thereforei j=|i||j|sin 90 k= (1)(1)(1)k=kExampleSuppose we want to findj i. Again, refer to Figure 5. If we apply the right hand screw rule,a vector perpendicular to bothjandi, in the sense defined by the right hand screw rule, is i= kExampleSuppose we want to findi i. Because these two vectors are parallel the angle between themis0 . We can use the Key Point developed on page 5 to show thati i= a similar manner we can derive all the results given in the following Key Point:Key Pointi i=0j j=0k k=0i j=kj k=ik i=jj i= kk j= ii k= mathcentre 2009We can use these results to develop a formula for finding the vector product of two vectors givenin cartesian form:Supposea=a1i+a2j+a3kandb=b1i+b2j+b3 kthena b= (a1i+a2j+a3k) (b1i+b2j+b3k)=a1i (b1i+b2j+b3k)+a2j (b1i+b2j+b3k)+a3k (b1i+b2j+b3k)=a1i b1i+a1i b2j+a1i b3k+a2j b1i+a2j b2j+a2j b3k+a3k b1i+a3k b2j+a3k b3k=a1b1i i+a1b2i j+a1b3i k+a2b1j i+a2b2j j+a2b3j k+a3b1k i+a3b2k j+a3b3k kNow, from the previous Key Point three of these terms are zero.

7 Those that are not zero simplifyto givea b= (a2b3 a3b2)i+ (a3b1 a1b3)j+ (a1b2 a2b1)kThis is the formula which we can use to calculate a vector product when we are given the cartesiancomponents of the two PointIfa=a1i+a2j+a3kandb=b1i+b2j+b3kthen a b= (a2b3 a3b2)i+ (a3b1 a1b3)j+ (a1b2 a2b1)kExampleSuppose we wish to find the vector product of the two vectorsa= 4i+3j+7kandb= 2i+5j+ use the previous result witha1= 4,a2= 3,a3= 7andb1= 2,b2= 5,b3= 4. Substitutioninto the formula givesa b= ((3)(4) (7)(5))i+ ((7)(2) (4)(4))j+ ((4)(5) (3)(2))kwhich simplifies toa b= 23i 2j+ mathcentre 2009 For those familiar with evaluation ofdeterminantsthere is a convenient way of rememberingand representing this formula which is given in the following Key Point and which is explained inthe accompanying video and in the Example PointIfa=a1i+a2j+a3kandb=b1i+b2j+b3kthen a b= ijka1a2a3b1b2b3 = a2a3b2b3 i a1a3b1b3 j+ a1a2b1b2 k= (a2 b3 a3 b2)i (a1 b3 a3 b1)j+ (a1 b2 a2 b1)ExampleSuppose we wish to find the vector product of the two vectorsa= 4i+3j+7kandb= 2i+5j+ write down a determinant, which is an array of numbers: in the first row we write the threeunit vectorsi,jandk.

8 In the second and third rows we write the three components ofaandbrespectively:a b= i j k4 3 72 5 4 We then consider the first element in the first row,i. Imagine covering up the elements in itsrow and column, to give the array 3 75 4 . This is a so-called2 2determinant and is evaluatedby finding the product of the elements on the leading diagonal(top left to bottom right) andsubtracting the product of the elements on the other diagonal (3 4 7 5 = 23). Theresulting number gives theicomponent of the final then consider the second element in the first row,j. Imagine covering up the elements inits row and column, to give the array 4 72 4 . This2 2determinant is evaluated, as before,by finding the product of the elements on the leading diagonal(top left to bottom right) andsubtracting the product of the elements on the other diagonal,(4 4 7 2 = 2). The resultis then multiplied by 1and this gives thejcomponent of the final answer, that is , we consider the third element in the first row,k.

9 Imagine covering up the elements in itsrow and column, to give the array 4 32 5 . This determinant is evaluated, as before, by mathcentre 2009the product of the elements on the leading diagonal (top leftto bottom right) and subtractingthe product of the elements on the other diagonal (4 5 3 2 = 14). The resulting numbergives thekcomponent of the final write all this as follows:a b= i j k4 3 72 5 4 = 3 75 4 i 4 72 4 j+ 4 32 5 k= (3 4 7 5)i (4 4 7 2)j+ (4 5 3 2)k= 23i 2j+ 14kExercises 11. Use the formulaa b= (a2b3 a3b2)i+ (a3b1 a1b3)j+ (a1b2 a2b1)kto find the vectorproducta bin each of the following cases.(a)a= 2i+ 3j,b= 2i+ 9j.(b)a= 4i 2j,b= 5i upon your Use the formula in Q1 to find the vector producta bin each of the following cases.(a)a= 5i+ 3j+ 4k,b= 2i 8j+ 9k.(b)a=i+j 12k,b= 2i+j+ Use determinants to find the vector productp qin each of the following cases.(a)p=i+ 4j+ 9k,q= 2i k.(b)p= 3i+j+k,q=i 2j For the vectorsp=i+j+k,q= i j kshow that, in this special case,p q=q For the vectorsa=i+ 2j+ 3k,b= 2i+ 3j+k,c= 7i+ 2j+k, show thata (b+c) = (a b) + (a c)5.

10 Some applications of the vector productIn this section we will look at some ways in which the vector product can be the vector product to find a vector perpendicular to twogiven of the common applications of the vector product is to finding a vector which is perpendicularto two given vectors. The two vectors should be non-zero and must not be we wish to find a vector which is perpendicular to bothof the vectorsa=i+ 3j 2kandb= 5i know from the definition of the vector product that the vectora bwill be perpendicularto bothaandb. So first of all we calculatea mathcentre 2009a b= i j k1 3 25 0 3 = (3 3 ( 2) 0)i (1 3 ( 2) 5)j+ (1 0 3 5)k= 9i 7j 15kThis vector is perpendicular occasions you may be asked to find a unit vector which is perpendicular to two given convert a vector into a unit vector in the same direction wemust divide it by its modulus of 9i 7j 15kis|a b|= ( 9)2+ ( 7)2+ ( 15)2= 355So, finally, the required unit vector is1 355( 9i 7j 15k).Using the vector product to find the area of a the parallelogram shown in Figure 6 which has sidesgiven by Figure 6.