高等数学第七章第三部分.ppt

第三节一、两向量的数量积二、两向量的向量积数量积向量积 1M 一、两向量的数量积沿与力夹角为?的直线移动,??W F 作用下, F 位移为 s , 则力 F所做的功为? cos sF ? 2Ms 启示两向量作这样的运算, 结果是一个数量. 1. 定义设向量的夹角为?,称记作数量积 ba?? cos ba的与为ba ba, 数量积也称为“点积”,“内积”?,0时当???a 上的投影为在ab ??记作故,0,时当同理???b a b ?jrPb b a ?jrP? cos b??baba a ?jrP??ba ba ? cos | |||baba ??????)2(??ba ????.ba ???)(?,0??ba ???,0||?a ?,0||?b ?,0 cos ???.ba ????.||)1( 2aaa ?????)(?,ba ????,0 cos ???.0 cos | |||????baba ????,0???.|| cos | ||| 2aaaaa ??????????证证??,2 ?,2 ???? 2. 性质向量先假设是非 0 3. 运算律(1) 交换律(2) 结合律),( 为实数?? abba?????ba)(?)(ba??)(ba???)()(ba?????)(ba????)(ba????(3) 分配律?? cbcacba??????0?c 时, 显然成立;时 )(ba? bab c ?jrP a c ?jrP ?? cba???? ba c??jrPc?c?? ??jrPjrP? a c ?jrP?cb c ?jrP?c ca??cb??)(jrPba c??AB C ?a bc 例1. 证明三角形余弦定理? cos 2 222ab bac???证:如图. 设,aBC?,bAC? cBA?则bac??? cos 2 222ab bac??? bac??? 2c)()(baba???aa??bb?? ba??2 2a? 2b?? cos bbaa???,, AB C ?a bc ,kajaiaa zyx???????kbjbibb zyx???????设??ba ??)(kajaia zyx?????)(kbjbib zyx?????? 4. 数量积的坐标表示)(iiba xx??)(jiba yx??)(kiba zx??)(ijba xy??)(kjba zy??)(ikba xz??)(jkba yz??)(jjba yy??)(kkba zz??,kji ??????,0???????ikkjji ??????,1||||||???kji ????.1???????kkjjii ?????? zzyyxxbabababa??????数量积的坐标表达式? cos | |||baba ??????