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Vector Part 1

# Vector Part 1 - Vectors Basic ideas 1 Basic idea of a...

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1 Vectors Basic ideas

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2 Basic idea of a vector Vector is something that needs both its size and its direction to completely describe it Example is velocity – how fast are we travelling and which direction are we travelling in Alternatives to vectors are scalars Scalars are things that are just described by a number ( e.g. temperature, density, mass)
3 Velocities as vectors These cars have the same speeds but different velocity vectors

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4 Situations where vectors occur Velocity vector - how fast and in what direction is earth travelling Force vector – points toward the sun in direction, size of force vector = strength of force
5 Force on a current in magnetic field Three vectors here - - the current vector J, - magnetic field vector B - F = the force vector on the wire

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6 Vectors needed in computer graphics There are two shiny spheres here with zebra stripes – program that made this picture needed to trace all the reflections by tracing light rays ( uses vectors in calculations )
7 Electrical charges

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8 History of vectors 1580 – Dutch engineer Stevins finds how to add forces as a vector triangle 1600-1800: People add and subtract vectors for forces and velocities Studying electricity and magnetism introduces more complicated problems – takes a long time to sort this out (Hamilton, Maxwell, Willard Gibbs - Gibbs found a good notation). They discovered two types of products involving vectors Internet demo of magnetic forces [trefil, hazen, student resource, chapter 17, ampere, quicktime]
9 Ordinary numbers as 1D vectors 1D = one dimensional Real numbers = positive numbers, negative numbers and zero Think of a positive number as a line segment pointing to the right; negative number points to left

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10 Ordinary numbers as 1D vectors Adding numbers gives idea for adding vectors Draw ‘ vector ’ for +1, Put vector for +2 at the end of +1 vector This gives sum as +3
11 Negative numbers Vector for -3 = Vector for +3 rotated by 180 degrees If we rotate -3 again we go back to +3 Motivates the idea that -( -3) = +3

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12 Notation OA Symbol used for a position vector is often OA – this is vector that starts at O and finishes at point A Other ways to write vectors use an arrow underneath or arrow above uuur uuur OA OA
13 Geometric vectors in 2D These vectors can be written as OA , BC , DE and FG

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14 Equal vectors Vectors BC and FG have the same lengths and same directions So as vectors they are considered the same We can write BC = FG Important to understand how to interpret this
15 Which vectors are equal here ?

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16 Adding vectors: OA + AB = OB 1. OA starts at O, goes to A 1. AB starts at A, goes to B 3. If we follow OA and then follow AB we end at B 4. OA + AB means follow OA , then follow AB
17 Heads and tails: OA + AB = OB This picture shows that adding vectors takes their sizes and directions into account Sometimes this is called the ‘head to tail method’

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Adding more than 2 vectors What is BC + CD ? What
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Vector Part 1 - Vectors Basic ideas 1 Basic idea of a...

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