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Vectors Wk 2 - Vectors Dot products and cross products of...

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Vectors Dot products and cross products of two vectors
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Two ways to multiply vectors We can add and subtract vectors Also can multiply a vector by a number There are two ways to ‘multiply’ vectors The dot product ‘multiplies’ two vectors and gives a number The vector product ‘multiplies’ two vectors and gives a vector There is no way to divide one vector by another vector
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The dot product (or scalar product) We have two vectors a and b The dot product is written a .b 1: explain where it comes from 2: then give the geometric meaning 3: Lastly get the coordinate version of a .b Applications include Getting angle between two vectors Calculating work done by a force Very useful in computer graphics calculations
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Pushing a cart along flat road Person pushes cart along the road – they must put in ‘work’ Two vectors F = force vector (person pushes parallel to road) d = displacement vector - this gives change in position F and d are parallel vectors Work = size of force needed * the distance pulled In this case Work = | F |. | d |
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More involved example Here a person is pushing at an angle to move a really large trolley The force vector is F The trolley moves along – its position changes by the vector d Person cannot push hard enough to lift the trolley so the vertical component of the force is not used The work to push the trolley depends on the horizontal component of the force and the distance travelled
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Formula for work needed when pushing a cart at angle Work = horizontal compt of F * distance travelled Work = | F |. cos( θ ). | d | = | F | .| d |. cos( θ ) This combination of vectors is an example of a dot product
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How much work is involved here ? The amount of work done is horizontal compt of force * distance moved Formula is | F | cos( θ ). | d | This is | F | . | d | cos( θ ).
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Summary This example was supposed to motivate the idea of considering the combination involving the size of two vectors and the cos(angle) between them. Now give the general geometric definition of the dot product
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Geometric definition of dot product a .b Use the smallest angle between a and b The definition of a .b = | a |. | b | cos( θ ) Note: a and b are both vectors The dot product is a number - it is not a vector
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Two special cases If a = b then θ = 0 get So a .a = | a |. |a | cos(0)
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Dot products of i and j in 2D i is a unit vector so its length is 1 | i | = 1 j is a unit vector and so | j | = 1 1: i . i = | i | 2 = 1 2 = 1 2: j . j = | j | 2 = 1 2 = 1 3: i . j = 0 because i and j are perpendicular
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Dot products of i , j and k in 3D i , j and k are all unit vectors So their lengths are 1 i . i = 1, j . j = 1 2 , k .k = 1 i. j and k are perpendicular so we have i . j = 0, i .k = 0 and j .k = 0
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Quick way to calculate dot product Geometric definition involves lengths and angles In 3D hard to calculate angles Need better method to get dot product Explain how to use component form of a and b to quickly get a .b
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Easier dot product formula in 3D It will always give the same answer as |a |.| b |.cos( θ )
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