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《全日制普通高级中学教科书(试验修订本·必修)数学第一册(下)》第106页给出了平面向量的基本定理:“如果e1、e2是同一平面内的两个不共线向量,那么对于这一平面内的任一向量α,有且只有一对实数λ1、λ2,使α=λ1e1+λ2e2·”那么如何求λ1、λ2呢?本文试图给出几种在解题时经常用到的方法,与同学们共同探讨. 一、直接法通过几何图形,由向量e1、e2出发求得向量α,从而求出实数λ1、λ2. 例1 如图1,在△OAB的边OA、OB上分别取M、N,使OM:OA=1:3,ON:OB=1:4,设线段AN和线段BM交于P点,且设OA=α,OB=b,若OP=ta+sb,求s、t的值.
The basic theorem of the plane vector is given in “Full-time Ordinary High School Textbooks (Experimental Revision, Compulsory) Mathematics, First Volume (Part 2)”, p. 106. “If e1 and e2 are two non-collinear vectors in the same plane. Then, for any vector α in this plane, there is only one pair of real numbers λ1 and λ2, so that α=λ1e1+λ2e2·” then how to find λ1 and λ2? This article tries to give several kinds of questions when solving problems. The method used, together with the students to discuss. First, the direct method through the geometry, starting from the vector e1, e2 obtained vector α, so as to find the real number λ1, λ2. Example 1 In Figure 1, on the side of the OAB OA Take M and N respectively on OB and OM: OA=1:3 and ON: OB=1:4. Set line segment AN and line segment BM to intersect at P, and set OA=α, OB=b, if OP= Ta+sb, find the value of s, t.