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

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1 1 Vector Mike Bailey vector.pptx Vector have Direction and Magnitude Magnitude: V V V V x y z 1

2 Vector Can lo Be Defined a the Poitional Difference Between Two Point 3 ( x, y, z ) ( P x, P y, P z ) ( V, V, V ) ( P, P, P) x y z x x y y z z Unit Vector have a Magnitude = V V V V V Vˆ V x y z The circumflex (^) tell u thi i a unit vector

3 Dot Product 5 ( x, y, z) B ( Bx, By, Bz) B( B B B ) B co x x y y z z Becaue it produce a calar reult (i.e., a ingle number), thi i alo called the Scalar Product Phyical Interpretation of the Dot Product 6 ˆB Thi i important memorize thi phrae! Bˆ co ^ = How much of live in the B direction 3

4 Phyical Interpretation of the Dot Product 7 F 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. Phyical Interpretation of the Dot Product F 8 The amount of the force accelerating the car along the road i how much of F i in the direction? F Fco Fˆ road 4

5 Phyical Interpretation of the Dot Product 9 F F Fco Fˆ road Dot Product are Commutative 10 B B Dot Product are Ditributive ( BC) ( B) ( C) 5

6 The Perpendicular to a D Vector 11 If V (x,y) then V ( y,x) You can tell that thi i true becaue V V (x,y) ( y,x) xy xy 0 co90 Cro Product 1 ( x, y, z) B ( Bx, By, Bz) B ( B B, B B, B B ) y z z y z x x z x y y x B B in Becaue it produce a vector reult (i.e., three number), thi i alo called the Vector Product 6

7 Phyical Interpretation of the Cro Product 13 ˆB Bˆ in Thi i important memorize thi phrae! ^ = How much of live perpendicular to the B direction The Perpendicular Property of the Cro Product 14 The vector B i both perpendicular to and perpendicular to B xb. The ight-hand-ule Property of the Cro Product Curl the finger of your right hand in the direction that tart at and head toward B. Your thumb point in the direction of xb. B 7

8 Cro Product are Not Commutative BB xb. Bx. 15 B B Cro Product are Ditributive ( BC) ( B) ( C) Ue for the Cro Product : Finding a Vector Perpendicular to a Plane (= the Surface Normal) 16 n S n( ) ( S) 8

9 Ue for the Cro Product : Finding a Vector Perpendicular to a Plane (= the Surface Normal) Thi i ued in CG Lighting 17 Ue for the Cro and Dot Product : I a Point Inide a Triangle? 3D (X-Y-Z) Verion 18 n S If P ( nn ),( nn ), and( nn ) Let: n( ) ( S) n ( ) ( P) q n ( S) ( P) r n ( S) ( PS) q r are all poitive, then P i inide the triangle S 9

10 n I a Point Inide a Triangle? Thi can be implified if you are in D (X-Y) S ES ( P) ( S) where: S ( S, S ) x x y y 19 P and: S ( S, S ) y y x x If ES, ES, E Similarly, ES ( PS) ( S) E ( P) ( ) are all poitive, then P i inide the triangle S Ue for the Cro Product : Finding the rea of a 3D Triangle 0 S height 1 rea BaeHeight Bae Height S in 1 1 rea S in ( ) ( S ) 10

11 S Derivation of the Law of Coine 1 r q ( ) ( ) [( S) ( S )] [( S) ( S )] [( S)( S)] [( S )( S )] ( S) ( S ) q r qr co S S Derivation of the Law of Sine r q * rea( S) ( S ) ( ) r in But, the area i the ame regardle of which two ide we ue to compute it, o: Dividing by (qr) give: r in q in qr in S in q in r in S 11

12 P d Ditance from a Point to a Plane nˆ In high chool, you defined a plane by: x + By + Cz + D = 0 3 It i more ueful to define it by a point on the plane combined with the plane normal vector 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 + By + Cz + D = 0 The perpendicular ditance from the point P to the plane i baed on the plane equation: dp nˆ The dot product i anwering the quetion How much of (P-) i in the direction?. Note that thi give a igned ditance. If d > 0., then P i on the ame ide of the plane a the normal point. Thi i very ueful. ˆn Where doe a line egment interect an infinite plane? 4 P 1 nˆ P The equation of the line egment i: P (1 t) P tp 0 1 If point P i in the plane, then: P,P,P,, (n,n,n ) 0 x y z x y z x y z If we ubtitute the parametric expreion for P into the plane equation, then the only thing we don t know in that equation i t. Solve 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 P lie inide a particular triangle. P 0 1

13 Minimal Ditance Between Two 3D Line 5 v p d v q P 0 0 The equation of the line are : P 0 P t v p 0 t v q The minimal ditance vector between the two line mut be perpendicular to both vector between them that i perpendicular to both i: v v p v q We need to anwer the quetion How much of ( 0 -P 0 ) i in the To do thi, we once again ue the dot product: d P0 0 vˆ v direction?. nother ue for Dot Product : Force One Vector to be Perpendicular to nother Vector 6 Here, we want to force to become perpendicular to B ˆB But, The trategy i to get rid of the parallel component, leaving jut the perpendicular ( Bˆ) Bˆ So that ( Bˆ) Bˆ Thi i known a Gram-Schmidt orthogonalization 13

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

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