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在向量数量积的定义 a·b=|a|·|b|cosθ中,当 b=a 时,得到了一个重要性质:a~2=|a|~2或|a|=a~2~(1/2).它最大的魅力就是实现了向量与数量之间的相互转化,从而我们可以将有关向量问题转化为数量问题来解决,又可以将有关数量问题转化为向量来解决.这样的转化,就像一个庞大的磁场,对许多数学问题产生了巨大的吸引力.一、求值例1 已知向量 a、b、c 满足 a+b+c=0,|a|=3,|b|=1,|c|=4,求 a·b+b·c+c·a 的值.解:由条件平方得(a+b+c)~2=0~2,展开得 a~2+b~2+c~2+2(a·b+b·c+c·a)=0,∴a·b+b·c+c·a
In the definition of the vector quantity product a·b=|a|·|b|cosθ, when b=a, an important property is obtained: a~2=|a|~2 or |a|=a~2~ (1/2). Its greatest attraction is the realization of the mutual transformation between vector and quantity, so that we can solve the vector problem by transforming it into a quantitative problem, and we can turn the related quantitative problem into a vector to solve it. Transformation, like a huge magnetic field, has huge appeal for many mathematical problems. First, the evaluation example 1 Known vectors a, b, c satisfy a+b+c=0,|a|=3,| b|=1,|c|=4, find the value of a·b+b·c+c·a. Solution: From the conditional squared (a+b+c)~2=0~2, expand a~ 2+b~2+c~2+2(a·b+b·c+c·a)=0, ∴a·b+b·c+c·a