Vector in the Two-dimensional Space(amath) @ iks84im62a的部落格

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Vector in the Two-dimensional Space(amath)

發問:

AD,BE and CF are the medians of triangle ABC ,G is the centroid,a,b,c,d,e,f and g are the position vectors of A,B,C,D ,E,F and G respectively.d=-i+3j,e=i-j and f=3i-2j (a) Prove that a+b+c+d=d+e+f (b)Find g (c) Find a,b and c Would you please give me the answer before 9 o'olock? 更新: Sorry to interrupt you. (a) should be a+b+c=d+e+f

最佳解答:

a. Since D is the mid-point of BC, thus d=(b+c)/2 Similarly, we have e=(a+c)/2 and f=(a+b)/2 Thus d+e+f = (b+c)/2 + (a+c)/2 + (a+b)/2 = a+b+c b. Since G is the centroid, we know that AG:GD=2:1 Thus g = (a+2d)/(1+2) = (a+2d)/3 = (a+b+c)/3 By (a), a+b+c = d+e+f, thus g = (a+b+c)/3 = (d+e+f)/3 = i Remark. This result means if you consider ΔDEF as a "mid-point triangle" of ΔABC, then they actually have the same centroid. c. Using the same argument in (a), we have d = (b+c)/2 e = (a+c)/2 f = (a+b)/2 Thus b+c = 2d = -2i+6j.........(1) a+c = 2e = 2i-2j............(2) a+b = 2f = 6i-4j............(3) a+b+c = 3g = 3i............(4) (4)-(1): a = 5i-6j (4)-(2): b = i+2j (4)-(3): c = -3i+4j

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