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dotProduct - The dot product in Rn 1 Dot product in Rn If x...

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The dot product in R n . 1. Dot product in R n : If x = ( x 1 , . . . , x n ) and y = ( y 1 , . . . , y n ) are points in R n then their dot product is defined by x · y = n k =1 x k y k . This is a natural extension to higher dimensions of the familiar “dot product” from elementary calculus. Observe that the dot product returns a scalar and not another point in R n . More precisely, the dot product is a function · : R n × R n R . It also can be checked that properties (a)-(d) listed below hold. Because its value is a real number, x · y is sometimes called the scalar product of x and y . Another important observation is that x · x = k x k 2 . The dot product has some basic properties: (a) x · x 0 with equality holding if and only if x = 0, (b) x · y = y · x , (c) x · ( y + z ) = x · y + x · z and (d) c ( x · y ) = ( cx ) · y = x · ( cy ) for any scalar c . 2. Projections in R n : If x and y are vectors from R n with y 6 = 0 then Proj y x , the projection of x along y , is defined as the vector in the direction of y such that x - Proj y x is orthogonal to y (cf., schematic below).
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