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Engineering Calculus Notes 62

Engineering Calculus Notes 62 - is the scalar −→ v ...

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50 CHAPTER 1. COORDINATES AND VECTORS Similar calculations for the y - and z -coordinates allow us to conclude that a 2 + b 2 c 2 = 2( x 1 x 2 + y 1 y 2 + z 1 z 2 ) and hence, substituting into Equation ( 1.16 ), factoring out 2, and recalling that a = | −→ v | and b = | −→ w | , we have | −→ v || −→ w | cos θ = x 1 x 2 + y 1 y 2 + z 1 z 2 . (1.17) This quantity, which is easily calculated from the entries of −→ v and −→ w (on the right) but has a useful geometric interpretation (on the left), is called the dot product 10 of −→ v and −→ w . Equation ( 1.17 ) appears already (with somewhat different notation) in Lagrange’s 1788 echanique Analitique [ 34 , N.11], and also as part of Hamilton’s definition (1847) of the product of quaternions [ 22 ], although the scalar product of vectors was apparently not formally identified until Wilson’s 1901 textbook [ 54 ], or more accurately Gibbs’ earlier (1881) notes on the subject [ 17 , p. 20]. Definition 1.4.1. Given any two vectors −→ v = ( x 1 ,y 1 ,z 1 ) and −→ w = ( x 2 ,y 2 ,z 2 ) in R 3 , their dot product
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Unformatted text preview: is the scalar −→ v · −→ w = x 1 x 2 + y 1 y 2 + z 1 z 2 . The de±nition of the dot product exhibits a number of algebraic properties, which we leave to you to verify (Exercise 3 ): Proposition 1.4.2. The dot product has the following algebraic properties: 1. It is commutative : −→ v · −→ w = −→ w · −→ v 2. It distributes over vector sums 11 : −→ u · ( −→ v + −→ w ) = −→ u · −→ v + −→ u · −→ w 3. it respects scalar multiples : ( r −→ v ) · −→ w = r ( −→ v · −→ w ) = −→ v · ( r −→ w ) . Also, the geometric interpretation of the dot product given by Equation ( 1.17 ) yields a number of geometric properties: 10 Also the scalar product , direct product , or inner product 11 In this formula, −→ u is an arbitrary vector, not necessarily of unit length....
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