V V The circumflex (^) tells us this is a unit vector

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Vector 1 Vector have Direction and Magnitude Mike ailey

pptx Vector Can lo e Defined a the oitional Difference

0 4 ( x, y, z ) ( x, y, z ) ( V, V, V ) (,, ) x y z x x y

1 Vector 1 Vector have Direction and Magnitude Mike ailey mjb@c.oregontate.edu Magnitude: V V V V x y z vector.pptx Vector Can lo e Defined a the oitional Difference etween Two oint 3 Unit Vector have a Magnitude = ( 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 circumflex (^) tell u thi i a unit vector 1

2 Dot roduct 5 hyical Interpretation of the Dot roduct 6 ( x, y, z) F ( x, y, z) ( ) co x x y y z z ecaue it produce a calar reult (i.e., a ingle number), thi i alo called the calar roduct The amount of the force accelerating the car along the road i how much of F i in the horizontal direction? F Fco road Thi i eay to ee in D, but a 3D verion of the ame problem i trickier. hyical Interpretation of the Dot roduct F 7 hyical Interpretation of the Dot roduct 8 F The amount of the force accelerating the car along the road i how much of F i in the direction? F Fco Fˆ road F Fco Fˆ road

3 Generalizing How Much of Live in the Direction 9 Generalizing How Much of Live erpendicular to the Direction 10 ˆ coθ coθ ˆ which i the length of the projection of onto the line ^ o, how much of live in the direction i that magnitude time the unit vector: From the previou lide, how much of live in the direction i : That, plu the perpendicular vector equal, o that how much of i perpendicular to the ^ direction i: ^ Dot roduct are Commutative 11 The erpendicular to a D Vector 1 If V (x,y) then V ( y,x) Dot roduct are Ditributive ( C) ( ) ( C) You can tell that thi i true becaue V V (x, y) ( y, x) xy xy 0 co90 3

4 Cro roduct 13 The erpendicular roperty of the Cro roduct 14 ( x, y, z) The vector i both perpendicular to and perpendicular to ( x, y, z) x. The ight-hand-ule roperty of the Cro roduct Curl the finger of your right hand in the direction that tart at and head toward. Your thumb point in the direction of x. (,, ) y z z y z x x z x y y x in ecaue it produce a vector reult (i.e., three number), thi i alo called the Vector roduct Cro roduct are Not Commutative 15 Ue for the Cro roduct : Finding a Vector erpendicular to a lane (= the urface Normal) 16 x. x. n n( ) ( ) Cro roduct are Ditributive ( C) ( ) ( C) 4

5 Ue for the Cro roduct : Finding a Vector erpendicular to a lane (= the urface Normal) Thi i ued in CG Lighting 17 n Ue for the Cro and Dot roduct : I a oint Inide a Triangle? 3D (X-Y-Z) Verion 18 Let: n( ) ( ) n ( ) ( ) q n ( ) ( ) r n ( ) ( ) If ( nn ),( nn ), and( nn ) q r are all poitive, then i inide the triangle I a oint Inide a Triangle? Thi can be implified if you are in D (X-Y) 19 Ue for the Cro roduct : Finding the rea of a 3D Triangle 0 n E ( ) ( ) where: and: (, ) imilarly, x x y y (, ) E ( ) ( ) E ( ) ( ) y y x x height 1 rea aeheight ae Height in If E, E, E are all poitive, then i inide the triangle 1 1 rea in ( ) ( ) 5

6 Derivation of the Law of Coine 1 Derivation of the Law of ine r r q ( ) ( ) q * rea( ) ( ) ( ) r in ut, the area i the ame regardle of which two ide we ue to compute it, o: [( ) ( )] [( ) ( )] [( )( )] [( )( )] ( ) ( ) q r qr co Dividing by (qr) give: r in q in qr in in in in q r d Ditance from a oint to a lane nˆ In high chool, you defined a plane by: x + y + Cz + D = 0 3 Where doe a line egment interect an infinite plane? 1 nˆ 4 It i more ueful to define it by a point on the plane combined with the plane normal vector The equation of the line egment i: (1 t) t 0 1 If you want the familiar equation of the plane, it i: x,y,z,, (n,n,n ) 0 x y z x y z which expand out to become the more familiar x + y + Cz + D = 0 The perpendicular ditance from the point to the plane i baed on the plane equation: d nˆ The dot product i anwering the quetion How much of (-) i in the direction?. Note that thi give a igned ditance. If d > 0., then i on the ame ide of the plane a the normal point. Thi i very 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 parametric expreion for into the plane equation, then the only thing we don t know in that equation i t. olve it for t*. Knowing t* will let u compute the (x,y,z) of the actual interection uing the line equation. If t* ha a zero in the denominator, then that tell u that t*=, and the line mut be parallel to the plane. Thi give u the point of interection with the infinite plane. We could now ue the method covered a few lide ago to ee if lie inide a particular triangle. 0 6

7 Minimal Ditance etween Two 3D Line 5 nother ue for Dot roduct : Force One Vector to be erpendicular to nother Vector 6 v p d v q Here, we want to force to become perpendicular to 0 0 The equation of the line are : 0 t v p 0 t The minimal ditance vector between the two line mut be perpendicular to both v q ˆ ut, The trategy i to get rid of the parallel component, leaving jut the perpendicular ( ˆ) ˆ vector between them that i perpendicular to both i: We need to anwer the quetion How much of ( 0-0 ) i in the v direction?. To do thi, we once again ue the dot product: d 0 0 vˆ v v p v q o that ( ˆ) ˆ Thi i known a Gram-chmidt orthogonalization 7

V V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics.

V V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics. 1 Vector Mike Bailey mjb@c.oregontate.edu vector.pptx Vector have Direction and Magnitude Magnitude: V V V V x y z 1 Vector Can lo Be Defined a the Poitional Difference Between Two Point 3 ( x, y, z )