Component Method

# Component Method - already been written suggestively to...

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Component Method The Pythagorean Theorem tells us that the length of a vector ( a , b , c ) is given by . This gives us a clue as to how we can define the dot product. For instance, if we want the dot product of a vector v = ( v 1 , v 2 , v 3 ) with itself ( v · v ) to give us information about the length of v , it makes sense to demand that it look like: v · v = v 1 v 1 + v 2 v 2 + v 3 v 3 Hence, the dot product of a vector with itself gives the vector's magnitude squared. Ok, that's what we wanted, but now a new question reigns: what is the dot product between two different vectors? The important thing to remember is that whatever we define the general rule to be, it must reduce to whenever we plug in two identical vectors. In fact, @@Equation @@ has
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Unformatted text preview: already been written suggestively to indicate that the general rule for the dot product between two vectors u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ) might be: u v = u 1 v 1 + u 2 v 2 + u 3 v 3 This equation is exactly the right formula for the dot product of two 3-dimensional vectors. (Note that the quantity obtained on the right is a scalar, even though we can no longer say it represents the length of either vector.) For 2-dimensional vectors, u = ( u 1 , u 2 ) and v = ( v 1 , v 2 ) , we have: u v = u 1 v 1 + u 2 v 2...
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## This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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