Coordinates and vectors. Background mathematics review

Transcription

1 Coordinates and vectors Background mathematics review David Miller

2 Coordinates and vectors Coordinate axes and vectors Background mathematics review David Miller

3 y Coordinate axes and vectors z x Ordinary geometry Three axes x, y, and z All at right angles Cartesian axes (from René Descartes) Lines or directions at right angles are also called orthogonal

4 y Coordinate axes and vectors z x Right-handed axes Using your right hand Thumb x Index ( first ) finger y Middle finger z No matter how you now rotate your whole hand the axes remain righthanded

5 y Coordinate axes and vectors z x If you use your left hand Thumb x Index ( first ) finger y Middle finger z give left-handed axes No rotation of this entire set of left-handed axes will ever make it right-handed We use right hand axes unless otherwise stated

6 z y P y origin. z P Coordinate axes and vectors For some point P in space. P The corresponding (x projections onto the P, y P, z P ) coordinate axes give x Cartesian coordinates P x x P, y P, and z P, relative to the origin of the axes Sometimes written (x P, y P, z P )

7 Coordinate axes and vectors G A vector is something with a magnitude such as a length and a direction Usually written in bold font e.g., G Sometimes G or G And shown as an arrow With length and direction

8 Coordinate axes and vectors A r B A vector could be the distance and direction you need to walk to get from A to B

9 Coordinate axes and vectors F A vector could be A force how hard you are pushing and what direction you are pushing

10 Coordinate axes and vectors v A vector could be A velocity how fast you are going (speed) e.g., the number on your car speedometer and what direction you are going in e.g., on a compass

11 r F v Coordinate axes and vectors An ordinary number which has no direction is called a scalar Distance how hard you push speed are all scalars Scalars are in ordinary fonts Usually italic in printing

12 y Coordinate axes and vectors G z G y k j i G G x x A vector has components along three orthogonal axes G x, G y, and G z We can also define vectors of unit length along each axis i unit vector along x j unit vector along y k unit vector along z z

13 y Coordinate axes and vectors G y Then we can write G=G x i+g y j+g z k G G y j G z G x i G x G z k x z

14 y Coordinate axes and vectors G y G G x i G y j G x x Then we can write G=G x i+g y j+g z k making the final vector up by adding its vector components G z k G z z

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16 Coordinates and vectors Operations with vectors Background mathematics review David Miller

17 Adding vectors G G + S S S G + S G To add vectors graphically connect them head to tail in any order

18 Adding vectors S z k G + S S y j G z k G y j G x i S x i To add vectors algebraically add them component by component G S G i G j G x x G S i G S j G S x y y S i S j x S z z k k y y z z k

19 Multiplying vectors Two kinds of multiplications or products for geometrical vectors Dot product ab Gives a scalar result Cross product a b Gives a vector result

20 Vector dot product angle a b One formula for the dot product is a b a b cos abcos Here the modulus sign means we take the length of the vector a a Note that a b b a Also 2 aa a So a aa

21 Vector dot product angle a b One formula for the dot product is a b a b cos abcos We can think of a b cos as The projection of vector b onto the direction of vector a Multiplied by the length of a or The projection of vector a onto the direction of vector b Multiplied by the length of b

22 Vector dot product a b One formula for the dot product is a b a b cos abcos Note that for two vectors at right angles /2 90 and cos / 2 0 so the dot product is zero

23 Vector dot product j k i 0 The unit vectors along the coordinate directions are all orthogonal (at right angles) So all their dots products with one another are zero i j ik 0 jk 0 ji 0 k i 0 k j 0 Also, since these are unit length vectors, by definition i i 1 jj 1 k k 1

24 Vector dot product a b Since i j 0 ik 0 jk 0 ji 0 k i 0 k j 0 Forming the dot product algebraically a b axi ayj azk bxi byj bzk gives a b axbx ayby azbz which is an equivalent formula for the dot product

25 Vector dot product G i The components of a vector can be found by taking the dot product with the unit vectors along the coordinate directions For example G i G i G y j G k i G x z x

26 Vector cross product For two vectors a a i a j a k x y z a b bxi byj bzk the vector cross product is a b n a b sin nabsin b n is a unit vector with a direction given by the right hand screw rule

27 a b a b gives vector n away from you b a a b gives vector n towards you Right hand screw rule Imagine you have a corkscrew With an ordinary right-handed thread with its handle lined up along vector a Now rotate the handle so it lines up with vector b The direction, in or out, that the corkscrew moved is the direction of the vector n

28 Vector cross product a b Note that a b b a If we have to turn clockwise to go from a to b So the corkscrew goes in So n points inwards Then we have to turn anti-clockwise to go from b to a So the corkscrew goes out So n point outwards

29 Vector cross product An equivalent algebraic formula for the vector cross product is a b a ybz azby i azbx axbz j axby aybx k A short-hand way of writing this is a b ax a y a z b b b x y z which is the same as the determinant notation used with matrix algebra i j k

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