# Vectors - VECTORS Many physical quantities have associated...

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1 VECTORS Many physical quantities have associated with them only a magnitude and so can be expressed by real numbers only. Some examples are temperature age mass density distance volume speed pressure. These numbers are called scalars . There are many quantities, however, that have associated with them both magnitude and direction, and these quantities cannot be described by a single real number. Some examples are velocity acceleration force displacement magnetic field current. To represent such quantities mathematically, we introduce vectors . For example, the velocity of a car is given as 90km/hr heading due North. Here 90km/hr is the speed of the car but it does not tell us in which direction the car is headed. The “due North” tells us in which direction the car is headed. A brief history of vectors 1580 Dutch engineer Stevins discovered how to add forces together to help design bridges and buildings. He didn’t use modern notation but he had the right idea. 1800’s Ampere and Michael Faraday found the force on an electric current in a magnetic field depended on the relative directions of the current and the magnetic field. The way to express this mathematically was not understood properly for nearly 30 years. 1844 Sir William Hamilton in Ireland is the first person to discover a way to “multiply” vectors. His approach, called quaternions, was rather complicated. 1880 American chemist Willard Gibbs found the notation that we use today. Today Gibbs is most famous for his work on thermodynamics but the good notation that he introduced for vectors is also a very important contribution.

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2 Simple examples where vectors are used. A plane is flying at a constant groundspeed of 1000 kilometres per hour due east and encounters a 90 kilometre per hour wind from the north west. Find the airspeed and direction you need to point the plane so that it continues to travel at the same speed and direction. ! v plane rel wind ! v wind ! ! v plane A ship is being pulled into position by two tug boats, as shown in the diagram below. If the tugs pull at constant but different forces, what angles do they each need to pull so that the ship continues to move in a straight line ? ! T 1 ! 1 ! 2 ! T 2
3 Geometric Vectors B A C D O E If A and B are two points in space, we denote AB ! " !! to be the vector from A to B. Similarly if D and E are two points in space, we denote DE ! " !! to be the vector from D to E. There are different notations for writing the vector from the point A to the point B: AB ! " !! an arrow above AB AB ! " !! an arrow below AB AB a line or squiggle under AB AB AB in bold type (in printed material). A vector from the origin to the point C is known as the position vector to C . You may wish to represent a vector by a single letter : for example the vector OC ! " !!

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## This note was uploaded on 04/14/2011 for the course MATHS 1201BPS taught by Professor Mariaaneiros during the Three '11 term at Griffith.

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Vectors - VECTORS Many physical quantities have associated...

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