21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl somethng straght dscussed n the precedng chapter. There s a relatonal operator 1 for vectors that allows us to bpass the calculaton of the moment arm. The relatonal operator s called the cross product. It s represented b the smbol read cross. The torque τ can be epressed as the cross product of the poston vector r for the pont of applcaton of the force, and the force vector F tself: τ =r F 21-1 efore we begn our mathematcal dscusson of what we mean b the cross product, a few words about the vector r are n order. It s mportant for ou to be able to dstngush between the poston vector r for the force, and the moment arm, so we present them below n one and the same dagram. We use the same eample that we have used before: s of Rotaton O Poston of the Pont of pplcaton of the Force F n whch we are loong drectl along the as of rotaton so t loos le a dot and the force les n a plane perpendcular to that as of rotaton. We use the dagramatc conventon that, the pont at whch the force s appled to the rgd bod s the pont at whch one end of the arrow n the dagram touches the rgd bod. Now we add the lne of acton of the force and the moment arm r to the dagram, as well as the poston vector r of the pont of applcaton of the force. 1 You are much more famlar wth relatonal operators then ou mght reale. The sgn s a relatonal operator for scalars numbers. The operaton s addton. pplng t to the numbers 2 and 3 elds 23=5. You are also famlar wth the relatonal operators,, and for subtracton, multplcaton, and dvson of scalars respectvel. 132
The Moment rm Lne of cton of the Force r O r Poston Vector for the Pont of pplcaton of the Force F The moment arm can actuall be defned n terms of the poston vector for the pont of applcaton of the force. Consder a tlted - coordnate sstem, havng an orgn on the as of rotaton, wth one as parallel to the lne of acton of the force and one as perpendcular to the lne of acton of the force. We label the as for perpendcular and the as for parallel. O r F 133
and Chapter 21 Vectors: The Cross Product & Torque Now we brea up the poston vector r nto ts component vectors along the aes. r O r r F From the dagram t s clear that the moment arm r s ust the magntude of the component vector, n the perpendcular-to-the-force drecton, of the poston vector of the pont of applcaton of the force. 134
Now let s dscuss the cross product n general terms. Consder two vectors, and that are nether parallel nor ant-parallel 2 to each other. Two such vectors defne a plane. Let that plane be the plane of the page and defne θ to be the smaller of the two angles between the two vectors when the vectors are drawn tal to tal. θ The magntude of the cross product vector s gven b = snθ The drecton of the cross product vector s gven b the rght-hand rule for the cross product of two vectors 3. To appl ths rght-hand rule, etend the fngers of our rght hand so that the are pontng drectl awa from our rght elbow. Etend our thumb so that t s at rght angles to our fngers. 21-2 2 Two vectors that are ant-parallel are n eact opposte drectons to each other. The angle between them s 180 degrees. nt-parallel vectors le along parallel lnes or along one and the same lne, but pont n opposte drectons. 3 You need to learn two rght-hand rules for ths course: the rght-hand rule for somethng curl somethng straght, and ths one, the rght-hand rule for the cross product of two vectors. 135
Keepng our fngers algned wth our forearm, pont our fngers n the drecton of the frst vector the one that appears before the n the mathematcal epresson for the cross product; e.g. the n. Now rotate our hand, as necessar, about an magnar as etendng along our forearm and along our mddle fnger, untl our hand s orented such that, f ou were to close our fngers, the would pont n the drecton of the second vector. Ths thumb s pontng straght out of the page, rght at ou! Your thumb s now pontng n the drecton of the cross product vector. C =. The cross product vector C s alwas perpendcular to both of the vectors that are n the cross product the and the n the case at hand. Hence, f ou draw them so that both of the vectors that are n the cross product are n the plane of the page, the cross product vector wll alwas be perpendcular to the page, ether straght nto the page, or straght out of the page. In the case at hand, t s straght out of the page. 136
When we use the cross product to calculate the torque due to a force F whose pont of applcaton has a poston vector r, relatve to the pont about whch we are calculatng the torque, we get an aal torque vector τ. To determne the sense of rotaton that such a torque vector would correspond to, about the as defned b the torque vector tself, we use The Rght Hand Rule For Somethng Curl Somethng Straght. Note that we are calculatng the torque wth respect to a pont rather than an as the as about whch the torque acts, comes out n the answer. Calculatng the Cross Product of Vectors that are Gven n,, Notaton Unt vectors allow for a straghtforward calculaton of the cross product of two vectors under even the most general crcumstances, e.g. crcumstances n whch each of the vectors s pontng n an arbtrar drecton n a three-dmensonal space. To tae advantage of the method, we need to now the cross product of the Cartesan coordnate as unt vectors,, and wth each other. Frst off, we should note that an vector crossed nto tself gves ero. Ths s evdent from equaton 21-2: = snθ, because f and are n the same drecton, then θ = 0, and snce sn 0 = 0, we have = 0. Regardng the unt vectors, ths means that: = 0 = 0 = 0 Net we note that the magntude of the cross product of two vectors that are perpendcular to each other s ust the ordnar product of the magntudes of the vectors. Ths s also evdent from equaton 21-2: = snθ, because f s perpendcular to then θ = 90 and sn 90 = 1 so = Now f and are unt vectors, then ther magntudes are both 1, so, the product of ther magntudes s also 1. Furthermore, the unt vectors,, and are all perpendcular to each other so the magntude of the cross product of an one of them wth an other one of them s the product of the two magntudes, that s, 1. 137
Now how about the drecton? Let s use the rght hand rule to get the drecton of : Fgure 1 Wth the fngers of the rght hand pontng drectl awa from the rght elbow, and n the same drecton as, the frst vector n to mae t so that f one were to close the fngers, the would pont n the same drecton as, the palm must be facng n the drecton. That beng the case, the etended thumb must be pontng n the drecton. Puttng the magntude the magntude of each unt vector s 1 and drecton nformaton together we see 4 that =. Smlarl: =, =, =, =, and =. One wa of rememberng ths s to wrte,, twce n successon:,,,,,. Then, crossng an one of the frst three vectors nto the vector mmedatel to ts rght elds the net vector to the rght. ut crossng an one of the last three vectors nto the vector 4 You ma have pced up on a bt of crcular reasonng here. Note that n Fgure 1, f we had chosen to have the as pont n the opposte drecton eepng and as shown then would be pontng n the drecton. In fact, havng chosen the and drectons, we defne the drecton as that drecton that maes =. Dong so forms what s referred to as a rght-handed coordnate sstem whch s, b conventon, the nd of coordnate sstem that we use n scence and mathematcs. If = then ou are dealng wth a left-handed coordnate sstem, somethng to be avoded. 138
139 mmedatel to ts left elds the negatve of the net vector to the left left-to-rght, but rght-to-left. Now we re read to loo at the general case. n vector can be epressed n terms of unt vectors: = Dong the same for a vector then allows us to wrte the cross product as: = Usng the dstrbutve rule for multplcaton we can wrte ths as: = = Usng, n each term, the commutatve rule and the assocatve rule for multplcaton we can wrte ths as: = Now we evaluate the cross product that appears n each term: 0 0 0 = Elmnatng the ero terms and groupng the terms wth together, the terms wth together, and the terms wth together elds:
140 = Factorng out the unt vectors elds: = whch can be wrtten on one lne as: = 21-3 Ths s our end result. We can arrve at ths result much more qucl f we borrow a tool from that branch of mathematcs nown as lnear algebra the mathematcs of matrces. We form the 33 matr b wrtng,, as the frst row, then the components of the frst vector that appears n the cross product as the second row, and fnall the components of the second vector that appears n the cross product as the last row. It turns out that the cross product s equal to the determnant of that matr. We use absolute value sgns on the entre matr to sgnf the determnant of the matr. So we have: = 21-4 To tae the determnant of a 33 matr ou wor our wa across the top row. For each element n that row ou tae the product of the elements along the dagonal that etends down and to the rght, mnus the product of the elements down and to the left; and ou add the three results one result for each element n the top row together. If there are no elements down and to the approprate sde, ou move over to the other sde of the matr see below to complete the dagonal.
For the frst element of the frst row, the, tae the product down and to the rght, ths elds mnus the product down and to the left the product down-and-to-the-left s. For the frst element n the frst row, we thus have: whch can be wrtten as:. Repeatng the process for the second and thrd elements n the frst row the and the we get and respectvel. ddng the three results, to form the determnant of the matr results n: = as we found before, the hard wa. 21-3 141