Cartesian vectors(20p) @ zogfly3的部落格

標題:

Cartesian vectors(20p)

發問:

figure1: http://wongju13.wix.com/chan (i) Show that vector a is perpendicular to vector c, but that b is not perpendicular to either a and c. (ii) Let w be a unit vector perpendicular to vectors a and c. Find w. (iii) Finf the projection of b onto the direction of w.

最佳解答:

a <-3,3,-2> b <0,1,-3> c <-4,-2,3> (i) if vectors are perpendicular, then their dot product = 0. so, a?c =<-3,3,-2>?<-4,-2,3> = (-3)(-4)+(3)(-2)+(-2)(3) = 12-6-6 = 0 but if it's b?a, then it's 9 which is not 0. and if it's b?c, it's -11, not 0. (ii) if it's perpendicular to a and c, then it's the normal vector to a and c. we should take cross product. so, n = a x c = <-3,3,-2> x <-4,-2,3> = <5,17,18> then, we need to find the unit vector. which is just divide the above normal vector by the magnitude of itself. and the magnitude of the normal vector n = sqrt(25+289+324) = sqrt(638) so, the unit vector is w = <5/sqrt(638), 17/sqrt(638), 18/sqrt(638)> (iii) projection b onto w. you should use the equation: [(b?w)/(||b||^2)] * w which is ([<0,1,-3>?<5/sqrt(638), 17/sqrt(638), 18/sqrt(638)>]/1) * <5/sqrt(638), 17/sqrt(638), 18/sqrt(638)>

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