Vecto Vecto have Diection and Magnitude Mike ailey mjb@c.oegontate.edu Magnitude: V V V V x y z vecto.pptx Vecto Can lo e Defined a the oitional Diffeence etween Two oint 3 Unit Vecto have a Magnitude =.0 4 ( x, y, z ( x, y, z ( V, V, V (,, x y z x x y y z z V V V V V Vˆ V x y z The cicumflex (^ tell u thi i a unit vecto Dot oduct 5 hyical Intepetation of the Dot oduct 6 ( x, y, z ( x, y, z ( co x x y y z z ecaue it poduce a cala eult (i.e., a ingle numbe, thi i alo called the cala oduct The amount of the foce acceleating the ca along the i how much of i in the hoizontal diection? co Thi i eay to ee in D, but a 3D veion of the ame poblem i tickie.
hyical Intepetation of the Dot oduct 7 hyical Intepetation of the Dot oduct 8 The amount of the foce acceleating the ca along the i how much of i in the diection? co ˆ co ˆ Genealizing How Much of Live in the Diection 9 Genealizing How Much of Live ependicula to the Diection 0 ˆ coθ coθ ˆ which i the length of the pojection of onto the line ^ o, how much of live in the diection i that magnitude time the unit vecto: om the peviou lide, how much of live in the diection i : ^ That, plu the pependicula vecto eual, o that how much of i pependicula to the ^ diection i: Dot oduct ae Commutative The ependicula to a D Vecto If V (x,y then V ( y,x Dot oduct ae Ditibutive ( C ( ( C You can tell that thi i tue becaue VV (x,y ( y,x xyxy 0 co90
Co oduct 3 The ependicula opety of the Co oduct 4 ( x, y, z The vecto i both pependicula to and pependicula to ( x, y, z x. The ight-hand-ule opety of the Co oduct Cul the finge of you ight hand in the diection that tat at and head towad. You thumb point in the diection of x. (,, y z z y z x x z x y y x in ecaue it poduce a vecto eult (i.e., thee numbe, thi i alo called the Vecto oduct Co oduct ae Not Commutative 5 Ue fo the Co oduct : inding a Vecto ependicula to a lane (= the uface Nomal 6 x. x. n n( ( Co oduct ae Ditibutive ( C ( ( C Ue fo the Co oduct : inding a Vecto ependicula to a lane (= the uface Nomal Thi i ued in CG Lighting 7 n Ue fo the Co and Dot oduct : I a oint Inide a Tiangle? 3D (X-Y-Z Veion 8 Let: n( ( n ( ( n ( ( n ( ( If ( nn,( nn, and( nn ae all poitive, then i inide the tiangle 3
n If I a oint Inide a Tiangle? Thi can be implified if you ae in D (X-Y E, E, E E ( ( whee: and: (, imilaly, ae all poitive, then i inide the tiangle x x y y (, E ( ( E ( ( y y x x 9 height Ue fo the Co oduct : inding the ea of a 3D Tiangle ea aeheight ae Height in ea in ( ( 0 Deivation of the Law of Coine Deivation of the Law of ine ( ( * ea( ( ( in ut, the aea i the ame egadle of which two ide we ue to compute it, o: [( ( ] [( ( ] [( ( ] [( ( ] ( ( co Dividing by ( give: in in in in in in d Ditance fom a oint to a lane nˆ In high chool, you defined a plane by: x + y + Cz + D = 0 3 Whee doe a line egment inteect an infinite plane? nˆ 4 It i moe ueful to define it by a point on the plane combined with the plane nomal vecto The euation of the line egment i: ( t t 0 If you want the familia euation of the plane, it i: x,y,z,, (n,n,n 0 x y z x y z which expand out to become the moe familia x + y + Cz + D = 0 The pependicula ditance fom the point to the plane i baed on the plane euation: d nˆ The dot poduct i anweing the uetion How much of (- i in the diection?. Note that thi give a igned ditance. If d > 0., then i on the ame ide of the plane a the nomal point. Thi i vey ueful. ˆn If point i in the plane, then:,,,, (n,n,n 0 x y z x y z x y z If we ubtitute the paametic expeion fo into the plane euation, then the only thing we don t know in that euation i t. olve it fo t*. Knowing t* will let u compute the (x,y,z of the actual inteection uing the line euation. If t* ha a zeo in the denominato, then that tell u that t*=, and the line mut be paallel to the plane. Thi give u the point of inteection with the infinite plane. We could now ue the method coveed a few lide ago to ee if lie inide a paticula tiangle. 0 4
Minimal Ditance etween Two 3D Line 5 nothe ue fo Dot oduct : oce One Vecto to be ependicula to nothe Vecto 6 v p d v Hee, we want to foce to become pependicula to 0 0 The euation of the line ae : 0 t v p 0 t The minimal ditance vecto between the two line mut be pependicula to both v ˆ ut, The tategy i to get id of the paallel component, leaving jut the pependicula ( ˆ ˆ vecto between them that i pependicula to both i: We need to anwe the uetion How much of ( 0-0 i in the v diection?. To do thi, we once again ue the dot poduct: d 0 0 vˆ v v p v o that ( ˆ ˆ Thi i known a Gam-chmidt othogonalization 5