1.3 dot-cross mixed product.ppt

1、Vector and Vector Space,Chapter 2,1.3 Multiplicative Operations on Vectors,1. The dot/scalar/inner product 2. The cross/vector product 3. The mixed product,Dot/scalar/inner product of two vectors点积/数量积/内积， Cross /outer/vector product of two vectors 叉积/外积/向量积, mixed product of three vectors混合积 Work 功

2、 Torque 转矩 Parallelogram 平行四边形 Parallelepiped 平行六面体,I、The dot/scalar/inner product of two vectors,1.Practical example,Remark,The result of this two vectors operation is a real value.,2. Definition of the dot product,Dot product also called as scalar product or inner product.,Conclusion: the dot prod

3、uct of two vectors is the multiplication of one of the two vectors moduli and the projection that another vector projects on it.,3.Properties of the dot product：,Proof:,Proof:,Property (2): are orthogonal if and only if,Particular ：,4.Operation law of the dot product：,（1）commutative law：,（2）distribu

4、tion law：,（3） ( )is a scalar：,let,coordinate representation for dot product,5.coordinate representation for dot product,6.Some application for coordinate representation of dot product, Find the norm of a vector,Example 1 Let,Solution,Find the included angle between two nonzero vectors,Then we have t

5、he sufficient and necessary condition for two vertical vector ., Find the projection,Solution:,2,Proof:,3,(P26-8),Exercise EP25-2, 3 , 5 , 6 , 7 ,8 , 9 (hint),II、 The cross product,1.An practical example,2.Definition,Supplement(geometric meaning of cross product ),3. Properties of cross product：,/,P

6、roof:,/,/,Particular ：,4.Operation rule：,（1）inverse commutative law:,（2）association law：,（3） is a scalar：,-Coordinate representation for cross product,5.Coordinate representation for cross product,let,cross product can be represented by 3-order determinant,/,So we can induce that,Note:,For example，,

7、Solution :,4,解,The area of triangle ABC is,Solution :,Exercise P25-4 ,6,III、The mixed product,Let,Coordinate representation of mixed product,（1）geometric significance of mixed product：,Some notes for mixed product：,(2) If the order of the three vectors keep the clockwise direction , and can exchange without change the mixed production value,2,3,1,Solution :,Example 6,7,Dot product,Cross product,Mixed product,