Vectors. February 1, 2010

Transcription

1 Vectors Febrary 1, 2010

2 Motivation Location of projector from crrent position and orientation: direction of projector distance to projector (direction, distance=magnitde) vector Examples: Force Velocity Displacement

3 Displacement Vectors Displacement vector = ordered pair of points, (A, B) Represented as an arrow A: tail, B: head Notation: (A, B) = AB = AB Magnitde: AB = AB. Direction: Geometric direction from A to B, if A B If A = B: Zero magnitde and non-specified direction (A, A) = AA = AA: zero displacement vector. Displacement vector with tail fixed at O: Position vector with respect to O; (O, P) = OP = OP = rp.

4 Eqality and Eqivalence of Displacement Vectors Eqality of displacement vectors: AB = DC (A, B) = (D, C) A = D and B = C Usefl for position vectors: OP = OQ P = Q. In general too restrictive. Eqal displacement vectors same magnitde and direction. Same magnitde and direction Eqal displacement vectors. Eqivalent displacement vectors Same magnitde and direction AB DC ABCD is a parallelogram.

5 Vectors Vector : set of displacement vectors with given direction and magnitde Magnitde of : common given magnitde. Direction of : common given direction, if non-zero magnitde. Set of zero displacement vectors = zero vector, 0. Representative for : displacement vector AB with the same direction and magnitde Intitive notation: = AB. Graphical representation: arrow withot fixed tail and head. Major advantage: we can translate displacement vectors. A C B D E F

6 By adding representative displacement vectors: Triangle Rle

7 By adding representative displacement vectors: Triangle Rle v B v C A + v Qv P + v R

8 By adding representative displacement vectors: Triangle Rle Properties:

9 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle

10 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle B v C B v C A A v D

11 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle Associative, ( + v) + w = + (v + w) Extends addition to + v + w

12 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle Associative, ( + v) + w = + (v + w) Extends addition to + v + w A B v + w v + v + v + w Figre: Sm of three vectors C w D

13 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle Associative, ( + v) + w = + (v + w) Extends addition to + v + w Opposite vector: If = AB, then AB + BA = 0, hence BA =.

14 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle Associative, ( + v) + w = + (v + w) Extends addition to + v + w Opposite vector: If = AB, then AB + BA = 0, hence BA =. Difference of vectors: v = v + : Parallelogram rle.

15 By adding representative displacement vectors: Triangle Rle Properties: Commtative, + v = v + : Parallelogram Rle A B v Associative, ( + v) + w = + (v + w) Extends addition to + v + w Opposite vector: If = AB, then AB + BA = 0, hence BA =. Difference of vectors: v = v + : Parallelogram rle. C A v B D v C

16 Linear Combinations Scalar mltiples: Let be a vector and c a real nmber (scalar) If c > 0 then c is the vector: with the same direction with magnitde c = c. If c < 0 then c = ( c)( ): opposite direction magnitde c = ( c)( ) = ( c) = c If c = 0 then c = 0. If c 1,..., c n are scalars and 1,..., n are vectors, then v = c c n n is the linear combination of given vectors with given scalars.

17 Decomposition of a vector along given directions Example: Tension indced by given force. F T 2 T 1 F 1 F 2 F

18 Vectors in Coordinates Oxyz: fixed rectanglar coordinate system i, j, k: nit vectors in the fndamental directions If P(a, b, c) is a point, then z ck P z (0, 0, c) P(a, b, c) x ai k O j i P x (a, 0, 0) bj P xy (a, b, 0) P y (0, b, 0) y OP = ai + bj + ck = a, b, c.

19 Operations in Coordinates Magnitde: a, b, c = OP = a 2 + b 2 + c 2 Addition: x 1, y 1, z 1 + x 2, y 2, z 2 = x 1 + x 2, y 1 + y 2, z 1 + z 2. Scalar mltiple: c x, y, z = cx, cy, cz. General displacement from A(x A, y A, z A ) to B(x B, y B, z B ): AB = AO + OB = OB OA = x B x A, y B y A, z B z A.